Question: The question is fairly simple and is actually stated in the title although I will elaborate a bit further on it. Also, I'm not really sure if this belongs here or it partly belongs to [cstheory.stackexchange.com](https://cstheory.stackexchange.com/).

Currently I'm trying to understand ZKP as a class of problems. I have read that IP=PSPACE and that ZKP exists under the assumption of OWF, and it basically is IP with the extra property of zero knowledge. I also have read that any NP problem has a ZKP, so NP $\subseteq$ ZKP. What do we know about IP and ZKP?

Edit : I haven't seen any classification diagrams with ZKP mentioned. Is there any resource available for this?

Answer: 
It is known that $\mathsf{IP} = \mathsf{PSPACE} = \mathsf{CZK}$ assuming the existence of one-way functions, see [here](https://dl.acm.org/doi/pdf/10.5555/88314.88333).

The existence of one-way functions is necessary, since it is already known that one-way functions are (essentially - there are minor caveats) necessary to prove that $\mathsf{CZK}$ contains $\mathsf{NP}$. See [here](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.58.8526&rep=rep1&type=pdf) and [here](https://people.seas.harvard.edu/%7Esalil/research/compzk-Apr06.pdf).

Of course, the first equality is for $\mathsf{CZK}$ with an *all powerful* prover. In cryptography, we are typically interested in zero-knowledge proofs where the prover is efficient given a witness. Having a witness for the language implies that the language is in $\mathsf{NP}$; it is known that, assuming one-way functions, all languages in $\mathsf{NP}$ have a zero-knowledge proofs with an *efficient* prover (see [here](https://people.csail.mit.edu/silvio/Selected%20Scientific%20Papers/Zero%20Knowledge/Proofs_That_Yield_Nothing_But_Their_Validity_or_All_Languages_in_NP_Have_Zero-Knowledge_Proof_Systems.pdf)), and as I said above, one-way functions are necessary for this.
Not sure, but I think I found the answer, I would really like to hear if this is actually correct.

The site [complexityzoo.net](https://complexityzoo.net/) mentions the [following](https://complexityzoo.net/Complexity_Zoo:C#czk) :

> 
> Assuming the existence of one-way functions, CZK contains NP [GMW91], and actually equals IP=PSPACE [[BGG+90]](https://dl.acm.org/doi/pdf/10.5555/88314.88333). However, none of these implications of one-way functions relativize (Impagliazzo, unpublished).
> 
> 
> 

I'm not really sure I can understand the last sentence, IP=ZK under the assumption of OWF.