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In a traditional Proof of Work (PoW) scheme it's assumed it takes some amount of time to compute the PoW function, such as SHA256x2 in Bitcoin and the difficulty increases as the network is capable of mining faster than the target 10 minutes.

In many Proof of Stake (PoS) schemes other metrics are used to determine if a miner was "selected" to have stake in the block, usually by looking at bits of TX inputs.

What keeps a PoS miner from having essentially infinite TX input hashes available on hand at any moment and mining blocks at a fast rate to inflate the currency / collect lots of subsidy rewards?

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There are many different proof of stake schemes, so let's just pick one to talk about. I will be talking about the one that was originally based in Peercoin and is now used by many other coins. And I will be talking about the latest version of that scheme as it evolved during the time from being very much insecure to be little more secure, but still way less secure than proof of work.

So the concept of difficulty remains there. If you mine blocks to fast, difficulty increases and you need more luck to be able to produce next block.

Another important characteristics is that the number of UTXOs does not matter that much. It helps you if you have many of them, but in order to be able to create next block, your chance is proportional to the total sum of the values of those UTXOs. Therefore imagine you have one UTXO with 1000 coins and someone as has 1000 UTXOs with just one coin. Your chance to create next block is the same, assuming we take just one round in isolation.

Only in long term due to other properties of the algorithm, it is slightly better to have many UTXOs. This is because once the UTXO is used to create a block, the new outputs of that transaction are forbidden to participate in block creation for some time.

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