Are there any proofs that BQP is not in P?
This is a famous open problem, so you won't find a proof anywhere at this time. We don't even know how to prove that $\mathsf{P\neq PSPACE}$, so finer separation of complexity classes such as $\mathsf{P\neq BQP}$ is also not known. Note that $\mathsf{BQP\subseteq PSPACE}$ is not trivial, you can find a proof in Quantum complexity theory, by Bernstein &amp; Vazirani.

It is suspected that they are different, with the ability of quantum computers to factor integers in polynomial time being a strong witness.