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It seems that the rationale behind the proof-of-work requirement in Bitcoin is that it creates trust in a decentralized trustless network by:

1. helping to reach consensus on which version of the blockchain is the correct one in case that there are multiple temporary competing forks.




2. making the accepted blockchain computationally immutable (irreversible).

Question: Why is proof-of-work required for creating trust?

If most users are honest, then they would voluntarily enforce the prohibition to rewrite the blockchain.

If most users are dishonest, they can still rewrite the blockchain, even with the proof-of-work requirement, not just by outperforming the honest miners, but even simply by creating a consensus to accept a new fork of the main blockchain as the correct version of the blockchain.

Proof of work does not create trust. It creates incentive.

Miners are paid if their block is eventually part of the main version of history ("blockchain") that the network accepts. They must irrecoverably burn electricity in order to create blocks, which costs them money; money they only get paid for if their block "wins".

These factors together mean that miners have a strong incentive to cooperate with other miners, instead of having each try to construct their own version of history, because ultimately only one version will be accepted (ignoring things like intentional forks in the network).

Proof of work is also used for a much more logistical, but equally important purpose: denial of service protection. Because the minimum difficulty for a block scales with the network's hashrate, it is enormously expensive for someone to produce more blocks than (on average) 1 per 10 minutes. Without an objective measure of work like PoW (or central trusted set of parties) anyone could spam the network with an unboundedly large amount of blocks to validate.

If most users are honest, then they would voluntarily enforce the prohibition to rewrite the blockchain.

Without proof of work, this is just not possible. There would be no limit to the number of blocks that could be created at about the same time, there could be thousands of them created every minute. Different servers might receive them in different orders. And a server that was down for a day would have no way to know what to accept.

You need some way to agree on which of two or more equally good ways the system can make forward progress will be agreed upon. Proof of work is not the only way to do this, but you need some way.

In addition, there would be no way to do the initial distribution of the bitcoins as they're created.

Proof of work is not required, there are other algorithms such as proof of stake that are still decentralized (though I don't know how it works even though Ethereum wants to use proof of stake mining). The proof of work gives a block a difficulty, saying "there must be at least X zeroes in a block." (X does not have to be an integer, if it did, difficulty could only be adjusted on orders of two, four, or sixteen; right now it is a little above 16.8) This adding zeroes controls the number of problems that can be solved, and is usually on the order of hundreds or (lower) tens of hundreds per day. When there are more miners, the network responds relatively soon by raising difficulty (in less than 2 weeks), and when there are less, it also responds by doing the opposite (in possibly more than 2 weeks). The proof of work is like a lottery, and who solves the problem is randomly selected and is then awarded the prize through the coinbase. Proof of work is basically a verification mechanism that looks at the zeroes in a hash (work) and forces a certain number (minimum) to be zero.