چه حکایت از فراقت که نداشتم ولیکن

The two notions are not specific to ZK, so it is in fact better to try to understand it in the context of general proof systems.

Soundness: the proof system is truthful, so whatever it proves, it is true.

Completeness: the proof system is comprehensive, so it can prove all true statements.

You can have completeness without soundness: if the proof system proves everything including falsity, obviously it is complete but not sound.

You can have soundness without completeness: if the proof system proves nothing, obviously it is sound (after all, it never proves any falsity), but not complete.

چه حکایت از فراقت که نداشتم ولیکن...

I can think of four major hurdles which are not entirely independent:NP-hardness only gives you information about complexity in the limit. For many NP-complete problems, algorithms exist that solve al چه حکایت از فراقت که نداشتم ولیکن...