We give a corrected proof that if PP \subseteq BQP/qpoly, then the Counting Hierarchy collapses, as originally claimed by [Aaronson 2006]. This recovers the related unconditional claim that PP does not have circuits of any fixed size n^k even with quantum advice. We do so by proving that YQP*, an oblivious version of (QMA \cap coQMA), is contained in APP, and so is PP-low.

Added exposition and corrected arithmetic errors in v2.

We give a corrected proof that if PP \subseteq BQP/qpoly, then the Counting Hierarchy collapses, as originally claimed by [Aaronson, CCC 2006]. This recovers the related unconditional claim that PP does not have circuits of any fixed size n^k even with quantum advice. We do so by proving that YQP*, an oblivious version of (QMA \cap coQMA), is contained in APP, and so is PP-low.