Amplifying circuit lower bounds against polynomial time, with applications

We give a self-reduction for the Circuit Evaluation problem, and prove the following consequences. \begin{itemize} \item {\bf Amplifying Size-Depth Lower Bounds.} If $\Circ Eval \in \SIZEDEPTH[n^k, n^{1-\delta}]$ for some $k$ and $\delta$, then for every $\eps > 0$, there is a $\delta' > 0$ such that $\Circ Eval \in \SIZEDEPTH[n^{1+\eps}, n^{1-\delta'}]$. Moreover, the resulting circuits require only $\tilde{O}(n^{\eps})$ bits of non-uniformity. As a consequence, %proving $\Circ Eval \not\in \SIZEDEPTH[n^{1.1}, n^{1-\delta}]$ for every $\delta > 0$ entails that ${\sf P} \not\subset \NC/\poly$. That is, strong enough depth lower bounds for Circuit Evaluation imply a full separation of $\P$ and $\NC$ (even with a weak size lower bound). \item {\bf Lower Bounds for Quantified Boolean Formulas.} Let $c, d > 1$ and $e