The main result in this paper is that any proof that $promise\textrm{-}\mathcal{BPP}=promise\textrm{-}\mathcal{P}$ necessitates proving that polynomial-sized circuits cannot simulate non-deterministic machines that run in arbitrarily small super-polynomial time (i.e., $NTIME[n^{f(n)}]\not\subseteq\mathcal{P}/\mathrm{poly}$, for essentially any $f(n)=\omega(1)$). The super-polynomial time bound in the conclusion of the foregoing conditional statement cannot be improved (to conclude that $\mathcal{NP}\not\subseteq\mathcal{P}/\mathrm{poly}$) without unconditionally proving that $\mathcal{P}\ne\mathcal{NP}$.

This paper is a direct follow-up to the very recent breakthrough of Murray and Williams (ECCC, 2017), in which they proved a new ``easy witness lemma'' for $NTIME[o(2^n)]$. Our main contribution is conceptual, in highlighting the strong ``barriers'' for proving $pr\mathcal{BPP}=pr\mathcal{P}$ (and even for proving much weaker statements) that can be demonstrated using their techniques. Following their approach, we apply the new lemma within the celebrated proof strategy of Williams (SICOMP, 2013), and derive the main result by using a parameter setting that is different than the ones they considered. We also include an alternative proof of the main theorem, which does not rely on the work of Murray and Williams, but rather uses a modification of the well-known lower bound of Santhanam (SICOMP, 2009).

* Added Sec 1.1 to discuss the meaning of the new barrier for proving prBPP=prP.

* Added a new and alternative proof of the main theorem (i.e., Thm 1).

* Minor revisions to the high-level exposition.

We show that any proof that $promise\textrm{-}\mathcal{BPP}=promise\textrm{-}\mathcal{P}$ necessitates proving circuit lower bounds that almost yield that $\mathcal{P}\ne\mathcal{NP}$. More accurately, we show that if $promise\textrm{-}\mathcal{BPP}=promise\textrm{-}\mathcal{P}$, then for essentially any super-constant function $f(n)=\omega(1)$ it holds that $NTIME[n^{f(n)}]\not\subseteq\mathcal{P}/\mathrm{poly}$. The conclusion of the foregoing conditional statement cannot be improved (to conclude that $\mathcal{NP}\not\subseteq\mathcal{P}/\mathrm{poly}$) without \emph{unconditionally} proving that $\mathcal{P}\ne\mathcal{NP}$.

This paper is a direct follow-up to the very recent breakthrough of Murray and Williams (ECCC, 2017), in which they proved a new ``easy witness lemma'' for $NTIME[o(2^n)]$. Following their approach, we apply the new lemma within the celebrated proof strategy of Williams (SICOMP, 2013), and derive our result by using a parameter setting that is different than the ones they considered.