> zk signature schemes exist and are implied by the existence of one way functions and publicly verifiable nizk.

Almost. The result is from CRYPTO89 paper of Bellare and Goldwasser. They derive a signature scheme from a nizk. It is not known whether you can get a nizk from a signature scheme. Moreover, no signature scheme can be a ZK: https://crypto.stackexchange.com/questions/35177/is-using-di...