What is a Non-Interactive Zero Knowledge Proof? I understand the concept of a Zero Knowledge Proof thanks to the easy to understand analogy of Alibaba's cave. However, this seems to require interaction between the verifier and the other party.

I have not found an explanation of non-interactive zero knowledge proofs (NIZK). The wikipedia article is way too complex for someone without advanced training to understand.

https://en.wikipedia.org/wiki/Non-interactive_zero-knowledge_proof

Can someone explain the concept of non-interactive zero-knowledge in a simple way?
A non-interactive ZK proof is when you play with yourself. Or, more accurately, with an impartial version of yourself.

In a normal ZK proof, the prover first issues a bunch of commitments, then the verifier issues challenges that the prover complies with; this proves anything only as long as the verifier is assumed to issue challenges normally without any prior understanding with the prover.

In a non-interactive ZK proof, the verifier is replaced by a hash function (or something similar) which is computed over the whole set of commitments: the hash function result is the challenge. If the hash function is really a random oracle then the prover cannot guess its output before trying it, i.e. before having produced his commitments, and that's where the security comes from.