We prove new results on the circuit complexity of Approximate Majority, which is the problem of computing Majority of a given bit string whose fraction of 1's is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and alternating time, \Sigma_{O(1)}Time (t). Our main results are the following:

1. We prove that (2^n^0.1)-size depth-3 circuits for Approximate Majority on n bits have bottom fan-in \Omega(log n). As a corollary we obtain that BPTime (t) \not \subseteq \Sigma_{2}Time (o(t^2)) with respect to some oracle. This complements the result that BPTime (t) \subseteq \Sigma_{2}Time (t^2 \cdot poly log t) with respect to every oracle (Sipser and Gacs, STOC '83; Lautemann, IPL '83).

2. We prove that Approximate Majority is computable by uniform polynomial-size circuits of depth 3. Prior to our work, the only known polynomial-size depth-3 circuits for Approximate Majority were non-uniform (Ajtai, Ann. Pure Appl. Logic '83). We also prove that BPTime (t) \subseteq \Sigma_{3}Time (t \cdot poly log t). This complements our results in (1).

3. We prove new lower bounds for solving QSAT_3 \in \Sigma_{3}Time (n \cdot poly log n) on probabilistic computational models. In particular, we prove that solving QSAT_3 requires time n^{1+\Omega(1)} on Turing machines with a random-access input tape and a sequentialaccess work tape that is initialized with random bits. No lower bound was previously known on this model (for a function computable in linear space).