# A query regarding relation b/w complexity classes $NP$ and $BQP$ in case of collapse of $PH$?

Assuming if $$PSPACE$$ collapses to the class $$NP^{NP}$$ $$\mathsf{\implies PH=\Sigma_2^P}=\mathsf{\Pi_2^P = PSPACE}$$.

Since $$\mathsf{NP\subseteq NP^{NP}}$$ and $$\mathsf{BQP\subseteq QMA \subseteq PSPACE} \implies \mathsf{BQP\subseteq NP^{NP}}$$.

Now, by definition:

$$NP$$ - class of languages verifiable in Polynomial time on a Non-Deterministic Turing machine.

$$BQP$$ - class of languages solvable in Polynomial time on a Quantum Computer.

Query: Does the above collapse implies any new relation (since none is currently known as of today) b/w the classes $$NP$$ and $$BQP$$ since now both belong to the same class $$NP^{NP}$$?

• I am baffalled at people in general seem to downvote the question without giving any indication why on this forum.. its very discouraging to even gather the will to ask a question here.. Sep 19 at 16:13
• I think people on this forum are sick and tired of questions of the form “if class $X$ collapsed to class $Y$, what would it mean for class $Z$?” They virtually never have any nonobvious answer. And, there’s a troll user who’s asked dozens of such things, exhausting everyone’s patience. Sep 19 at 19:28
• @J.Doe I am baffled but usually such collapse, if you state with oracles (which I think is meaningless and superficial yet glorified and popularized as important evidence for class separations by the community as a whole), would hold water at the site.
– Mr.
Sep 20 at 14:49