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Assume that there is an algorithm that ensures: 1- That a full node chooses an honest random wait time. 2- That the same node has waited the specified amount of time before proceeding to mine a block.

Would there be any need for mining using PoW and Why? Note that I said an algorithm and not a central entity like Intel Software Guard Extensions (SGX).

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Keeping block time at 10 minutes is only one of the many benefits that PoW provides. Other important ones include preventing double spending and transaction censorship. Most of the PoW alternative consensus mechanisms out there simply fail to ensure this level of security. There is an interesting conversation on the Bitcoin mailing list that dives into Proof-of-Stake and its inefficiencies in ensuring the above.

Double Spending: PoW ensures that a node has spent electricity and compute power in order to find a hash that matches the current target. This ensures that there is a cost involved in case any one wants to 'rewind' the blocks and make a separate transaction that double spends the previous one. This would also mean, that the person who wants to double spend would need to amass more hash power (and hence investment) than rest of the network combined, an almost impossible feat.

Transaction censorship: In cases where you do not have PoW, an attacker will bootstrap millions of nodes to ensure that he is the one that gets chosen to form the block. In that scenario, he can ensure that certain transactions do not enter the blockchain, aka censorship. Censorship attacks cannot be prevented except by ensuring that no single entity can claim 51% control of new block creation. By ensuring this, we can ensure that at least some other entities are unlikely to keep a transaction out of the blockchain, and in particular that no entity can make a short-ranged history rewrite if a censored transaction does get into the blockchain from the efforts of another block producer. This is trivial under proof-of-work, and is ensured by asking people to make huge investment to get the hash power needed to mine block. Since it is non-trivial to acquire energy PoW prevents censorship attacks.

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