We show that if the Boolean hierarchy collapses to level k, then the polynomial hierarchy collapses to BH3(k), where BH3(k) is the kth level of the Boolean hierarchy over P 2 . This is an improvement over the known results [3], which show that the polynomial hierarchy would collapse to PNPNP [O(log n)] . This result is significant in two ways. First, the theorem says that a deeper collapse of the Boolean hierarchy implies a deeper collapse of the polynomial hierarchy. Also, this result points to some previously unexplored connections between the Boolean and query hierarchies of P 2 and P 3 . Namely, BH(k) = co-BH(k) = BH3(k) = co-BH3(k) PNP [k] = PNP [k+1] = PNPNP [k+1] = PNPNP [k+2] . Key words: polynomial time hierarchy, Boolean hierarchy, polynomial time Turing reductions, oracle access, nonuniform algorithms, sparse sets AMS (MOS) subject classifications: 68Q15, 03D15, 03D20