Is the Presburger arithmetic decision problem known to be outside of BQP or BPP?
Let $~{\mathrm{PRESARITH}}$ denote the decision problem of the truth of statements in Presburger Arithmetic. As you note, [Fischer+Rabin&nbsp;1974] (PS manuscript) show that the nondeterministic time complexity of $\mathrm{PRESARITH}$ is at least $2^{2^{\Omega(n)}}$. In particular, $\mathrm{PRESARITH} \notin \mathsf{NEXP}$.
However, we have the containments
$$ \mathsf{
BPP \;\subseteq\; BQP \;\subseteq\; PP \;\subseteq\; PSPACE \;\subseteq\; EXP \;\subseteq\; NEXP
}
$$

Thus, it directly follows that $\mathrm{PRESARITH} \notin \mathsf{BQP}$, and in particular it is not in $\mathsf{BPP}$ either.