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 SIGMAtwo language,
acceptable by a SIGMAtwo-machine
with running time O(n^{k^2+k}),
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.
We also take this opportunity to publish
some unpublshed older results of the first author on
constant depth circuits,
both straight lower bounds and inapproximability results
based on decision tree type Switching Lemmas.
(This is a revised version of C-161.)