We consider the problem of proving circuit lower bounds
against the polynomial-time hierarchy.
We give both positive and negative results.
For the positive side,
for any fixed integer k > 0,
we give an explicit Sigma^p_2 language,
acceptable by a Sigma^p_2-machine
with running time O(n^{k^2}log^{k+1}n),
that requires circuit size $>n^k$.
This provides a constructive version of an existence theorem of Kannan.
Our main theorem is on the negative side.
We give evidence that
it is infeasible to give relativizable proofs
that any single language in the polynomial-time hierarchy requires
super polynomial circuit size.
Our proof techniques are based on
the decision tree version of the Switching Lemma
for constant depth circuits
and Nisan-Wigderson pseudorandom generator.
(This is a revised version of C-161 (May 2002).)