The Boolean Hierarchy and the Polynomial Hierarchy: A Closer Connection

We show that if the Boolean hierarchy collapses to level $k$, then the polynomial hierarchy collapses to $BH_{3}(k)$, where $BH_{3}(k)$ is the $k$th level of the Boolean hierarchy over $\Sigma^{P}_{2}$. This is an improvement over the known results, which show that the polynomial hierarchy would collapse to $P^{NP^{NP}[O(\log n)]}$. 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 $Delta^{P}_{2}$ and $Delta^{P}_{3}$. Namely, \begin{eqnarray*} & BH(k) = {\rm co-}BH(k) \implies BH_{3}(k) = {\rm co-}BH_{3}(k), \\[\jot] & P^{{\rm NP}\|[k]} = P^{{\rm NP}\|[k+1]} \implies P^{{\rm NP}^{\rm NP}\|[k+1]} = P^{{\rm NP}^{\rm NP}\|[k+2]}. \end{eqnarray*}