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You often hear people say that Bitcoin is a pseudonymous system, and that it's not anonymous. What is the actual difference in the context of electronic money?

Pseudonymity means that you can see the payment history of a pseudonym (eg an address), but there's no way (in the system itself) to link the pseudonym to a person.

What would an anonymous system look like? I am aware of Chaumian e-cash (https://bitcoin.stackexchange.com/a/10666/69518), but what properties must anonymous electronic cash have? Is it what Andrew Poelstra alludes to:

Total anonymity, in the sense that when you spend money there is no trace of where it came from or where it's going, is theoretically possible by using the cryptographic technique of zero-knowledge proofs.

So you can see a payment, but you can't see where the money is coming from or where it's going (just like chaumian e-cash). Would anything less than this be considered not anonymous? I'm not looking for solutions, only what properties are needed for an electronic money system to be anonymous.

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Would anything less than this be considered not anonymous?

That depends who is doing the considering.

I suggest thinking about degrees of anonymity rather than absolutes such as anonymous vs psedonymous vs identifiable. There are systems which claim to offer more anonymity. There may be trade-offs in other aspects of security or in other factors such as convenience.

what properties are needed for an electronic money system to be anonymous.

What reduces anonymity in Bitcoin is the property you already identified. The public nature of the record of addresses used in transactions. So removing that property would increase the degree of anonymity. An example might be the gradual transition in usage of transaction types from ones that explicitly include a public key in the locking script towards types with locking scripts that only disclose something like a hash of a script? I don't know if the issue is fundamentally tied to the UTXO model though.

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