What is complexity theoretic implication of following possib

What is complexity theoretic implication of following possibilities - NP=coNP=BPP=RP or coNP?NP=BPP=RP (consensus is these seem impossible)?

Solution

Unless I\'m missing something obvious (it is getting late here), the second isn\'t possible - BPP=coBPP, so NP=BPP?NP=coBPP?NP=coNP.

In the first case, some of the consequences would be:

\"Feasible\" algorithms for NP-complete problems, for at least the randomised version of feasible.
PH?NP
Unless P=NP, adding randomness adds computational power.
Unless P=NP, strong pseudorandom number generators do not exist.