# Difference between BPP, IP and AM complexity?

I just can't figure out the difference between these three classes of complexity, does anyone know the difference and can explain it in a simple, direct way without too many definitions involved? Thank you very much!

• For those interested in Proof Assistants, there is a new proposed SE site ProofAssistants Nov 28 '21 at 8:35

All of these classes can be described as consisting of languages accepted by certain Arthur–Merlin games. In these games, Merlin, an unlimited but potentially dishonest party, tries to convince Arthur, a probabilistic Turing machine, that the input $$x$$ belongs to the language $$L$$. We ask that if $$x \in L$$ then some Merlin convinces Arthur with probability at least $$2/3$$, and if $$x \notin L$$ then every Merlin convinces Arthur with probability at most $$1/3$$.
• AM protocols have two rounds, without loss of generality. IP protocols have polynomially many rounds; the number of rounds could depend on $n$. I suggest ignoring the technical issue of public vs private coins for now. Oct 11 '21 at 21:54