What are some examples of pairs of complexity classes $A$ and $B$ such thatwe do not know whether $A=B$, andwe do not know contradictory relativizations either (i.e., we do not know oracles $P$ and $Q$ such that $A^P = B^P$ and $A^Q \ne B^Q$)?To phrase the question another way, what are some exceptions to the heuristic that if can't figure out contradictory relativizations then it is easy to resolve the equality question outright?
Potentially equal complexity classes without known contradictory relativizations
cc.complexity theory;complexity classes;relativization
I think the biggest such example at present is $BQP $ (quantum polybomial time) vs $PH $ (the polynomial time hierarchy). Significant effort has been put into separating them relative to an oracle, with no success. (Of course a powerful enough oracle will make them equal.) And the best known containment result is that $BQP $ is in $PP $. Some references for attacks on the oracle problem:http://arxiv.org/abs/0910.4698http://arxiv.org/abs/1007.0305