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pengvado comments on Quantum Russian Roulette - Less Wrong
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P⊆NP⊆PP⊆PSPACE, but we don't even have a proof of P≠PSPACE.
Unconditionally proven strict containments are pretty rare. You'll probably want to settle for some lesser evidence.
Hah. Thanks, that settles the issue for now: short of Scott Aaronson making an unexpected breakthrough, we have no proof that quantum suicide computing can solve any non-polynomial problems :-)