Let f(n) be a space-constructible superpolynomial function. Then BQP $\subseteq$ PSPACE $\subset$ SPACE(f(n)), so in particular, SPACE(f(n)) $\not\subseteq$ BQP. Let L be a problem such that every problem in SPACE(f(n)) is BQP-reducible to L. Then L $\notin BQP$\notin$ BQP.

Are there any problems that have been proven to not be in BQP for which that is not known to be provable by the above method?

Let f(n) be a space-constructible superpolynomial function. Then BQP $\subseteq$ PSPACE $\subset$ SPACE(f(n)), so in particular, SPACE(f(n)) $\not\subseteq$ BQP. Let L be a problem such that every problem in SPACE(f(n)) is BQP-reducible to L. Then L $\notin BQP$.

Are there any problems that have been proven to not be in BQP for which that is not known to be provable by the above method?