Particularly, we already have some oracle separation results such as $\mathbf{BPP}^A\neq \mathbf{BQP}^A$ [Simon], $\mathbf{NP}^A\not\subseteq \mathbf{BQP}^A$ [BBBV], and $\mathbf{BQP}^A\not\subseteq \mathbf{NP}^A$ [Bernstein and Vazirani]. But given that most problems are non-relativising, how does such oracle separation even counted as evidence of unconditional separation? Or do we need to discuss it case-wisely?
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It can hardly be considered evidence for inequality or equality. We know $\mathsf{IP} = \mathsf{PSPACE}$, but there is an oracle $A$ relative to which $\mathsf{IP}^A \neq \mathsf{PSPACE}^A$ (as proved here). Similarly, there are classes which are not equal, even though their relativized versions to a certain oracle are equal (see here for a couple examples).
The reason for all this is that, for a complexity class $\mathsf{C}$ (corresponding to a machine model which admits oracles) and an oracle $O$, $\mathsf{C}^O$ is potentially a different class altogether from $\mathsf{C}$—and it is probably best to treat it as such.
dkaeae
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At best, the evidence given is only heuristic and informal, but it is still important. Oracles in the examples you gave do address the general question: how does quantum computing compare to nondeterminism and randomness, in power? The oracles definitely do not answer the original unrelativized questions, rather they provide a different (related) black-box model where the question provably has a negative answer. Since unconditional lower bounds are very difficult to come by, oracles can be useful information, but again it's only heuristic.
Ryan Williams
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