As I understand the schnorr authentication protocol is an interactive protocol where a prover WLOG proves to a Verifier that he possesses the discrete log to some group element.
If we use the Fiat-shamir heuristic and turn this into a non-interactive protocol, then this becomes the schnorr signature scheme.
Is the following correct:
Since Schnorr signature scheme is a signature scheme, we would like it to be unforgable, non-repudiation, etc; all the properties of a signature scheme.
Since the Schnorr authentication protocol is an interactive protocol, we need it to be complete and sound.
Our Schnorr Signature scheme does not need to be sound or complete because it does not make sense for a signature scheme.
If the non-interactive version of the schnorr authentication protocol was not a signature scheme, then we would need it to be sound and complete.