Earlier version:

#### Abstract

In several settings derandomization is known to follow from circuit
lower bounds that themselves are equivalent to the existence of
pseudorandom generators. This leaves open the question whether
derandomization implies the circuit lower bounds that are known to
imply it, i.e., whether the ability to derandomize in *any* way
implies the ability to do so in the canonical way through pseudorandom
generators.
For the setting of decision problems, Impagliazzo et al. implicitly
showed the following equivalence: Randomized polynomial-time decision
procedures can be simulated in NSUBEXP (the subexponential
version of NP) with subpolynomial advice on infinitely many input
lengths if *and only if* NEXP is not in P/poly.
We establish a full analogue in the setting of verification
procedures: Arthur-Merlin games can be simulated in
Σ_{2}SUBEXP (the subexponential version of
Σ_{2}P)
with subpolynomial advice on infinitely many input lengths if
*and only if* Σ_{2}EXP is not in NP/poly.

A key ingredient in our proofs is improved Karp-Lipton style collapse
results for nondeterministic circuits. The following are two instantiations
that may be of independent interest: Assuming that Arthur-Merlin games can
be derandomized in Σ_{2}P, we show that

- if PSPACE is contained in NP/poly then PSPACE collapses to
Σ
_{2}P, and
- if coNP is contained in NP/poly then PH collapses to
P^Σ
_{2}P.