If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.

Polynomials

Polynomials are a special class of algebraic expressions of the form:

• x+5
• x^4
• x^3+3x^2+3x+1
• 628x^{271}+318x^{270}+530x^{269}+…+69x+381

i.e. they are a sum of any (finite!) number of terms of the form cx^k

There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.

Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:

• A(0)+B(0)=C(0)
• A(1)+B(1)=C(1)
• A(2)+B(2)=C(2)
• A(3)+B(3)=C(3)

And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:

• 12+1=13
• 10+8=18
• 15+8=23
• 15+13=28

You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:

• A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
• B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
• C(x)=5x+13

Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.

Comparing a polynomial to itself

You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)

As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.

Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).

Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!

Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.

The metaverse is the most commonly used term in current technology discussions. While the entire tech ecosystem awaits the metaverse's full arrival, defining it is difficult. Imagine the internet in the '80s! The metaverse is a three-dimensional virtual world where users can interact with digital solutions and each other as digital avatars.
The metaverse is a three-dimensional virtual world where users can interact with digital solutions and each other as digital avatars.

Among the metaverse hype, the Decentraland vs Sandbox debate has gained traction. Both are decentralized metaverse platforms with no central authority. So, what's the difference and which is better? Let us examine the distinctions between Decentraland and Sandbox.

2 Popular Metaverse Platforms Explained

The first step in comparing sandbox and Decentraland is to outline the definitions. Anyone keeping up with the metaverse news has heard of the two current leaders. Both have many similarities, but also many differences. Let us start with defining both platforms to see if there is a winner.

Decentraland

Decentraland, a fully immersive and engaging 3D metaverse, launched in 2017. It allows players to buy land while exploring the vast virtual universe. Decentraland offers a wide range of activities for its visitors, including games, casinos, galleries, and concerts. It is currently the longest-running metaverse project.

Decentraland began with a $24 million ICO and went public in 2020. The platform's virtual real estate parcels allow users to create a variety of experiences. MANA and LAND are two distinct tokens associated with Decentraland. MANA is the platform's native ERC-20 token, and users can burn MANA to get LAND, which is ERC-721 compliant. The MANA coin can be used to buy avatars, wearables, products, and names on Decentraland.

Sandbox

Sandbox, the next major player, began as a blockchain-based virtual world in 2011 and migrated to a 3D gaming platform in 2017. The virtual world allows users to create, play, own, and monetize their virtual experiences. Sandbox aims to empower artists, creators, and players in the blockchain community to customize the platform. Sandbox gives the ideal means for unleashing creativity in the development of the modern gaming ecosystem.

The project combines NFTs and DAOs to empower a growing community of gamers. A new play-to-earn model helps users grow as gamers and creators. The platform offers a utility token, SAND, which is required for all transactions.

What are the key points from both metaverse definitions to compare Decentraland vs sandbox?

It is ideal for individuals, businesses, and creators seeking new artistic, entertainment, and business opportunities. It is one of the rapidly growing Decentralized Autonomous Organization projects. Holders of MANA tokens also control the Decentraland domain.

Sandbox, on the other hand, is a blockchain-based virtual world that runs on the native token SAND. On the platform, users can create, sell, and buy digital assets and experiences, enabling blockchain-based gaming. Sandbox focuses on user-generated content and building an ecosystem of developers.

Sandbox vs. Decentraland

If you try to find what is better Sandbox or Decentraland, then you might struggle with only the basic definitions. Both are metaverse platforms offering immersive 3D experiences. Users can freely create, buy, sell, and trade digital assets. However, both have significant differences, especially in MANA vs SAND.

For starters, MANA has a market cap of $5,736,097,349 versus $4,528,715,461, giving Decentraland an advantage.
The MANA vs SAND pricing comparison is also noteworthy. A SAND is currently worth $3664, while a MANA is worth $2452.

The value of the native tokens and the market capitalization of the two metaverse platforms are not enough to make a choice. Let us compare Sandbox vs Decentraland based on the following factors.

Workstyle

The way Decentraland and Sandbox work is one of the main comparisons. From a distance, they both appear to work the same way. But there's a lot more to learn about both platforms' workings. Decentraland has 90,601 digital parcels of land.

Individual parcels of virtual real estate or estates with multiple parcels of land are assembled. It also has districts with similar themes and plazas, which are non-tradeable parcels owned by the community. It has three token types: MANA, LAND, and WEAR.

Sandbox has 166,464 plots of virtual land that can be grouped into estates. Estates are owned by one person, while districts are owned by two or more people. The Sandbox metaverse has four token types: SAND, GAMES, LAND, and ASSETS.

Age

The maturity of metaverse projects is also a factor in the debate. Decentraland is clearly the winner in terms of maturity. It was the first solution to create a 3D blockchain metaverse. Decentraland made the first working proof of concept public. However, Sandbox has only made an Alpha version available to the public.

Backing

The MANA vs SAND comparison would also include support for both platforms. Digital Currency Group, FBG Capital, and CoinFund are all supporters of Decentraland. It has also partnered with Polygon, the South Korean government, Cyberpunk, and Samsung.

SoftBank, a Japanese multinational conglomerate focused on investment management, is another major backer. Sandbox has the backing of one of the world's largest investment firms, as well as Slack and Uber.

Compatibility

Wallet compatibility is an important factor in comparing the two metaverse platforms. Decentraland currently has a competitive advantage. How? Both projects' marketplaces accept ERC-20 wallets. However, Decentraland has recently improved by bridging with Walletconnect. So it can let Polygon users join Decentraland.

Scalability

Because Sandbox and Decentraland use the Ethereum blockchain, scalability is an issue. Both platforms' scalability is constrained by volatile tokens and high gas fees. So, scalability issues can hinder large-scale adoption of both metaverse platforms.

Buying Land

Decentraland vs Sandbox comparisons often include virtual real estate. However, the ability to buy virtual land on both platforms defines the user experience and differentiates them. In this case, Sandbox offers better options for users to buy virtual land by combining OpenSea and Sandbox. In fact, Decentraland users can only buy from the MANA marketplace.

Innovation

The rate of development distinguishes Sandbox and Decentraland. Both platforms have been developing rapidly new features. However, Sandbox wins by adopting Polygon NFT layer 2 solutions, which consume almost 100 times less energy than Ethereum.

Collaborations

The platforms' collaborations are the key to determining "which is better Sandbox or Decentraland." Adoption of metaverse platforms like the two in question can be boosted by association with reputable brands. Among the partners are Atari, Cyberpunk, and Polygon. Rather, Sandbox has partnered with well-known brands like OpenSea, CryptoKitties, The Walking Dead, Snoop Dogg, and others.

Platform Adaptivity

Another key feature that distinguishes Sandbox and Decentraland is the ease of use. Sandbox clearly wins in terms of platform access. It allows easy access via social media, email, or a Metamask wallet. However, Decentraland requires a wallet connection.

Prospects

The future development plans also play a big role in defining Sandbox vs Decentraland. Sandbox's future development plans include bringing the platform to mobile devices. This includes consoles like PlayStation and Xbox. By the end of 2023, the platform expects to have around 5000 games.

Decentraland, on the other hand, has no set plan. In fact, the team defines the decisions that appear to have value. They plan to add celebrities, creators, and brands soon, along with NFT ads and drops.

Final Words

The comparison of Decentraland vs Sandbox provides a balanced view of both platforms. You can see how difficult it is to determine which decentralized metaverse is better now. Sandbox is still in Alpha, whereas Decentraland has a working proof of concept.

Sandbox, on the other hand, has better graphics and is backed by some big names. But both have a long way to go in the larger decentralized metaverse. 

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Musk’s purchase of Twitter has sealed his bond with the American right—whether the platform’s left-leaning employees and users like it or not.

Elon Musk's pursuit of Twitter Inc. began earlier this month as a joke. It started slowly, then spiraled out of control, culminating on April 25 with the world's richest man agreeing to spend $44 billion on one of the most politically significant technology companies ever. There have been bigger financial acquisitions, but Twitter's significance has always outpaced its balance sheet. This is a unique Silicon Valley deal.

To recap: Musk announced in early April that he had bought a stake in Twitter, citing the company's alleged suppression of free speech. His complaints were vague, relying heavily on the dog whistles of the ultra-right. A week later, he announced he'd buy the company for $54.20 per share, four days after initially pledging to join Twitter's board. Twitter's directors noticed the 420 reference as well, and responded with a “shareholder rights” plan (i.e., a poison pill) that included a 420 joke.

Musk - Patrick Pleul/Getty Images

No one knew if the bid was genuine. Musk's Twitter plans seemed implausible or insincere. In a tweet, he referred to automated accounts that use his name to promote cryptocurrency. He enraged his prospective employees by suggesting that Twitter's San Francisco headquarters be turned into a homeless shelter, renaming the company Titter, and expressing solidarity with his growing conservative fan base. “The woke mind virus is making Netflix unwatchable,” he tweeted on April 19.

But Musk got funding, and after a frantic weekend of negotiations, Twitter said yes. Unlike most buyouts, Musk will personally fund the deal, putting up up to $21 billion in cash and borrowing another $12.5 billion against his Tesla stock.

Free Speech and Partisanship

Percentage of respondents who agree with the following

The deal is expected to replatform accounts that were banned by Twitter for harassing others, spreading misinformation, or inciting violence, such as former President Donald Trump's account. As a result, Musk is at odds with his own left-leaning employees, users, and advertisers, who would prefer more content moderation rather than less.

Dorsey - Photographer: Joe Raedle/Getty Images

Previously, the company's leadership had similar issues. Founder Jack Dorsey stepped down last year amid concerns about slowing growth and product development, as well as his dual role as CEO of payments processor Block Inc. Compared to Musk, a father of seven who already runs four companies (besides Tesla and SpaceX), Dorsey is laser-focused.

Musk's motivation to buy Twitter may be political. Affirming the American far right with $44 billion spent on “free speech” Right-wing activists have promoted a series of competing upstart Twitter competitors—Parler, Gettr, and Trump's own effort, Truth Social—since Trump was banned from major social media platforms for encouraging rioters at the US Capitol on Jan. 6, 2021. But Musk can give them a social network with lax content moderation and a real user base. Trump said he wouldn't return to Twitter after the deal was announced, but he wouldn't be the first to do so.

Trump - Eli Hiller/Bloomberg

Conservative activists and lawmakers are already ecstatic. “A great day for free speech in America,” said Missouri Republican Josh Hawley. The day the deal was announced, Tucker Carlson opened his nightly Fox show with a 10-minute laudatory monologue. “The single biggest political development since Donald Trump's election in 2016,” he gushed over Musk.

But Musk's supporters and detractors misunderstand how much his business interests influence his political ideology. He marketed Tesla's cars as carbon-saving machines that were faster and cooler than gas-powered luxury cars during George W. Bush's presidency. Musk gained a huge following among wealthy environmentalists who reserved hundreds of thousands of Tesla sedans years before they were made during Barack Obama's presidency. Musk in the Trump era advocated for a carbon tax, but he also fought local officials (and his own workers) over Covid rules that slowed the reopening of his Bay Area factory.

Teslas at the Las Vegas Convention Center Loop Central Station in April 2021. The Las Vegas Convention Center Loop was Musk's first commercial project. Ethan Miller/Getty Images

Musk's rightward shift matched the rise of the nationalist-populist right and the desire to serve a growing EV market. In 2019, he unveiled the Cybertruck, a Tesla pickup, and in 2018, he announced plans to manufacture it at a new plant outside Austin. In 2021, he decided to move Tesla's headquarters there, citing California's "land of over-regulation." After Ford and General Motors beat him to the electric truck market, Musk reframed Tesla as a company for pickup-driving dudes.

Similarly, his purchase of Twitter will be entwined with his other business interests. Tesla has a factory in China and is friendly with Beijing. This could be seen as a conflict of interest when Musk's Twitter decides how to treat Chinese-backed disinformation, as Amazon.com Inc. founder Jeff Bezos noted.

Musk has focused on Twitter's product and social impact, but the company's biggest challenges are financial: Either increase cash flow or cut costs to comfortably service his new debt. Even if Musk can't do that, he can still benefit from the deal. He has recently used the increased attention to promote other business interests: Boring has hyperloops and Neuralink brain implants on the way, Musk tweeted. Remember Tesla's long-promised robotaxis!

Musk may be comfortable saying he has no expectation of profit because it benefits his other businesses. At the TED conference on April 14, Musk insisted that his interest in Twitter was solely charitable. “I don't care about money.”

The rockets and weed jokes make it easy to see Musk as unique—and his crazy buyout will undoubtedly add to that narrative. However, he is a megabillionaire who is risking a small amount of money (approximately 13% of his net worth) to gain potentially enormous influence. Musk makes everything seem new, but this is a rehash of an old media story.