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This isn't my area, so many apologies if I say something incorrect:
1) "What evidence do we have that $\mathsf{BPP}\subseteq \mathsf{REXP}$?"
Isn't this unconditionally true? It should follow from $\mathsf{BPP}\subseteq \mathsf{EXP}$, and in fact by Sipser-Gács-Lautemann $\mathsf{BPP}\subseteq \Sigma^p_2\cap \Pi_2^p$.
2) "What consequences does $\mathsf{...
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There are proposals for quantum money where it appears that not even the bank can produce two copies of a quantum money state with the same serial number. See
Farhi et al's paper Quantum Money from Knots,
Mark Zhandry's paper Quantum Lightning Never Strikes the Same State Twice,
Daniel Kane's paper Quantum Money from Modular Forms.
In order to make ...
3
Yes, even in the purely existential case. See https://en.wikipedia.org/wiki/Existential_theory_of_the_reals
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