Theorem: If NP is in BPP then the whole polynomial-time hierarchy is in BPP.
Let's focus on simply showing Σ2 is in BPP if NP is in BPP. The rest is straightforward induction. Here is our first proof:
To get a correct proof (first due to Zachos) we need to use Arthur Merlin games. Consider a Σ2 language L as an ∃∀ expression. Since NP is in BPP, we can replace the ∀ with a probabilistic test. This gives us what is known as MA or a Merlin-Arthur game where the powerful Merlin sends a message that Arthur can probabilistically verify. A now classic result shows that MA is contained in AM, where Arthur provides a random string to Merlin who must then provide a proof based on that string. Once again we apply the NP in BPP assumption to allow Arthur to simulate Merlin probabilistically and now we have a BPP algorithm for L.
The problem in the first proof is in the second "⊆". The assumption NP in BPP does not imply NPA in BPPA for all A.
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