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In PoW several nodes compete to be elected. If n1 does not get elected eventually nX will be elected. So PoW does clearly not need such a concept of timeslots.

In PoS lets say a node gets elected via random election. If that node is byzantine we clearly need a mechanism to achieve liveness. Every PoS algorithm that I've seen so far does introduce the concept of time-slots.

1) How do nodes reach consensus about the current time?

2) How do these PoS algorithms which use time-slots ensure that the time-source is not centralized?

3) How do these PoS algorithms which use time-slots ensure that the time-source is in sync?

4) How do you come to a consensus if a block was delivered in time? I mean even two non-byzantine nodes cannot possibly come to a consensus if an elected producer produced a block in time. Maybe n1 (honest) received the block in time and n2 (honest) has not received the block in time.

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Honestly, this question pertains more to Cryptography as a research field than to operational IT. I could only give you a lead on where to start:

A long blog post by Vitalik: https://medium.com/@VitalikButerin/a-proof-of-stake-design-philosophy-506585978d51

Yellow-paper: http://gavwood.com/paper.pdf

Would also recommend searching for anything done by Vlad Zamfir as he was the project leader for the development of Casper (POS).

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