Are zero-knowledge proofs quantum-resistant?
There is no generic answer to this question. Zero-knowledge proof (ZKP) systems can be post quantum secure, but they need not be. It all depends on the cryptography on which the security of the ZKP rests.
There exist ZKPs that only use collision-resistant hash functions and hence are plausibly post quantum secure. Popular examples include Hyrax, Aurora, or zkSTARK.
Other notable ZKPs like Bulletproofs or libsnark rely on cryptographic assumptions which are known to be vulnerable to quantum attacks (e.g. hardness of discrete log in certain elliptic curve groups).
Informally, post-quantum cryptosystems strive to remain secure even in the scenario that the attacker has a large quantum computer. Precisely, the basic hard assumptions such as DLP, BDH and so on are not secure in this scenario.
Regarding the Zero-Knowledge proof systems, it depends on the setup phase and model of working in each ZK proof scheme. For instance, ZK-SNARKs are based on a trusted setup which is executed by a third trusted party. At this point, we suppose these schemes are secure because there is no adversary to find a solution for some hard problems in the polynomial time. So a quantum attacker is able to compromise the security by attempting to find a solution for the claimed hard problem. Each user who has the master key is able to do some malicious activities. I should emphasize that if the setup phase in this schemes is based on some quantum-resistant hard assumptions so this scheme will resist against this kind of attack.
Conversely, in the ZK-STARKs, there is no third trusted party to produce secure parameters and the prover and verifier just use some plain parameters which are based on pure mathematics instead of a master key. So in this model, there is no secure parameter with some assumptions based on hard problems. You can take a look at this post. LINK