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How will these storage coins stop cheaters? Why can't I claim that I will store your data and just not do it?
Check out their whitepaper https://filecoin.io/filecoin.pdf

It includes two proofs, proof-of-replication and proof-of-spacetime which help prove someone is storing the file multiple times in independent locations.

It looks like some kind of zero knowledge proof. Why I can't just generate that and then discard your data?
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Because you need to keep proving over time
Reading the paper I get the idea. Although it's possible to cheat, nodes have to put filecoins at risk of loss if they get caught.
Amazing, although I am not sure I understand their business model correctly. So they are a middle man for digital storage rental and those transactions are done in Filecoins that only they can mine and distribute? So that company has >$200M in cash now or who gets that money? And the investors are in because they saw the surge of Bitcoin (and now of Filecoin)?
Who is putting up this kind of cash?!
Accredited investors ;-) Didn't notice the difference with non accredited ones.