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What are some of them? I'm not too up on the latest research in this area so it would be interesting to know how you can do a cryptocurrency that doesn't depend on proof-of-work calculations.

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Proof of stake. ETH has an ongoing project called Casper to switch to PoS.

PoS has many of its own problems, but I think it is undeniably better from a waste point of view.

The issue with PoS is that it introduces trust into the element. There doesn't seem to be a way (yet?) where you can introduce PoS without lowering the security of new adopters.

How does PoS introduce trust into the system?

PoS works on the premise that rewards will be given to holders. Unlike PoW, PoS has a single incentive structure where the management of the networks is aligned with the holders of the tokens. If you are not a large holder, the existing holders can re-write the rules to their advantage. You have to trust that large holders won't do this, even though the incentives align with that outcome.

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