Strong Exponential Time Hypothesis: no less than O(2^n) algorithm for SAT (proved only with blackbox model)
Quantum Strong Exponential Time Hypothesis: no less than O(2^{n/2}) quantum algorithm for SAT (proved only with blackbox model)
Grover: solve SAT with high probability using 2^{n/2}c instruction given AND/OR/NOT
circuit C with length c.
Bernstein-Vazirani (XOR): O(c) with 100\%. (while classical need O(nc)), verifiable output, unverifiable input
Simon's Algorithm: not a natural problem, unverifiable input
Quantum Factoring: polynomial (while classical need O(\exp(n^{1/3}))), verifiable
Decision Factoring: whether there exist a factor of F in range [2, k]
when decision factoring return "Yes": verifiable
when decision factoring return "No": still verifiable (unlike SAT)
If we assume NP \neq coNP, then \text{NP-Complete} problems should not have such "no-witness". And therefore, factoring is not \text{NP-Complete}.
Bias-Busting: if code C biased then Pr\{0...0\} > 0, else Pr\{0...0\} = 0.
SAT \leq_p \text{Bias-Busting} (so it is NP-Hard)
If satisfiable, then C must be biased. If unsatisfiable, then C must be unbiased.
But Pr\{\} > 0 is too weak to solve NP-Hard problems.
\text{Bias-Busting} probably not in NP, since otherwise, a witness will be similar to Bias-Busting itself.
Toda-Ogiwara Theorem: Bias-Busting not in NP assuming P^{\Sigma_2} \neq NP^{\Sigma_2}
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