Is PoW stateless and without progress?

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I was listening a lecture on PoW, and there was a discussion about the pros of PoW, and one of the pros named was absence of state and progress. But now I’m thinking, is it really so, when in fact you’re progressing through a search space and you should have at least some state to represent a counter iterating through this search space? So, you kind of progressing? Or maybe I got the semantics wrong?

Btw lecture is awesome

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