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What would happen to transactions on Monero in a post-quantum world? Would all of the transactions become transparent to the 3/4 letter agencies?

Does the 'mixing' of coins happen on a server (so you trust the server not to log anything) or does it happen p2p so said agencies can analyze the network?

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I'm not sure that anyone can speak to what happens to crypto in a post quantum world.

Mixing does not happen on a server - that would be an atrocious violation of privacy. Monero (and other cryptonote coins ) use ring signatures - https://lab.getmonero.org/pubs/MRL-0004.pdf

There is also work being done to employ gmaxwell's Confidential Transactions: https://github.com/ShenNoether/MiniNero/blob/master/RingCT0....

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