I have some basic questions about complexity theory that came up when I tried to understand the result by Raz and Tal that BQP$ ^O\nsubseteq$ PH$ ^O$ . Aaronsons paper was helpful, but I still have some questions left.

Raz and Tal derive Corollary 1.5 from 1.4 “by the relation between blackbox separations and oracle separations”, but aren’t those the same thing? I thought the following all mean the same:
 blackbox model
 oracle model
 relative/relativized problem
 query complexity
It would make more sense to me if Corollary 1.5 follows from 1.4 by the relation between promise problems and decision problems in the blackbox model.

I am not sure how to interpret the fact that there is randomness in the problem. I think of a class such as BPP as languages solvable by a regular Turing machine with access to a random tape, which can make an error with a constant probability over its random bits. However we need to consider the worst case instantiation of the problem (right?). For a language to be in PH, there needs to be a PHmachine (without access to randomness) that does not fail on any input. Now for any oracleoutput coming from one distribution there is a nonzero probability that it was actually generated by the other distribution, so the PHmachine will be wrong sometimes. Why is that argument not sufficient to show that the problem is not in PH (or any other “zeroerror” class for that matter)?

What exactly is meant by an AC$ ^0$ circuit with access to an oracle? I’ve seen this described as “a circuit with access to the oracle’s truth table”, which I could understand if the oracle solved a decision problem $ f: \{0,1\}^n \rightarrow \{0,1\}$ , then the circuit input nodes are $ x_i = f(i)$ for $ i \leq 2^n$ . However, here the oracle samples from one of two distributions on $ \{\pm 1\}^{2N}$ and the circuit is defined as $ A: \{\pm 1\}^{2N} \rightarrow \{\pm 1\}$ . Does that mean the circuit only gets access to a single query output?