In Zero-Knowledge (ZK) proofs/arguments of knowledge, the ZK knowledge property informally says that it is possible to simulate the output of a (malicious) verifier interacting with a prover knowing a witness $w$ without using $w$ at all:
$$\{OUT_{V^*_\lambda} \langle P(w,x) , V^*_\lambda(s_\lambda, x) \rangle\}_{\lambda,x,w} \approx_c \{Sim(x, V^*_\lambda, s_\lambda)\}_{\lambda,x,w}$$
where $\lambda$ is a security parameter, $w$ is a witness of $x$ and $s_\lambda$ a non-uniform advice.
However this seems to be too weak for my use-case, as the malicious verifier has a (partial) access to the witness $w$.
Is there a stronger notion of ZK that works for cases where parts of the witness leak to the verifier? If yes, do we have protocols that achieve this stronger notion? (ideally post-quantum secure) What is the status in the non-interactive case (NIZK)?
I just saw in this course a notion of adaptively-secure NIZK that seems quite close to my needs, but it only focuses on NIZK, seems to be even stronger as the instance $(x,w)$ can be maliciously chosen by the adversary (in my case the instance is chosen uniformly at random by the prover) and don't say if there exists protocols achieving this notion.
