$\mathsf{TC}^0$ is a small class with $\oplus\mathsf{L}$ containing it.
Following inclusions are known: $$\mathsf{Cuniform}\mbox{ -}\mathsf{ACC}^0\subseteq\mathsf{Cuniform}\mbox{ -}\mathsf{TC}^0\subseteq\mathsf{Cuniform}\mbox{ -}\oplus\mathsf{L}\subseteq\mathsf{P}$$ $$\mathsf{PP}\not\subseteq\mathsf{DLOGTIMEuniform}\mbox{ -}\mathsf{TC}^0$$ $$\mathsf{DeterminantMod2}\in\oplus\mathsf{L}$$ where $\mathsf{C}\in\{\mathsf{P},\mathsf{LOGSPACE},\mathsf{DLOGTIME}\}$ and where $\mathsf{DeterminantMod2}$ is the problem of finding determinant of a $0/1$ square matrix modulo $2$.
It is possible $\mathsf{Permanent}$ could reduce to $\mathsf{P}$ by reduction via problems in $\mathsf{DeterminantMod2}$.
So how bad of collapse can we prove to avoid so far?
Is it known
- $$\mathsf{CH}\not\subseteq\mathsf{DLOGTIMEuniform}\mbox{ -}\mathsf{TC}^{{0}^{^{^{\mathsf{C'uniform}\mbox{ -}\oplus\mathsf{L}}}}}$$ and/or
- $$\mathsf{CH}\not\subseteq\mathsf{DLOGTIMEuniform}\mbox{ -}\mathsf{ACC}^{{0}^{^{^{\mathsf{C'uniform}\mbox{ -}\mathsf{NC}^1}}}}$$ for any $\mathsf{C'}\in\{\mathsf{P},\mathsf{LOGSPACE},\mathsf{DLOGTIME}\}$?
In other words is it known there cannot be a $\mathsf{DLOGTIMEuniform}\mbox{ -}\mathsf{ACC}^0$ algorithm utilizing $\mathsf{DLOGTIMEuniform}\mbox{ -}\mathsf{NC}^1$ oracle which computes the permanent?
It is known http://cjtcs.cs.uchicago.edu/articles/1999/7/contents.html provides $$\mathsf{PP}\not\subseteq\mathsf{DLOGTIMEuniform}\mbox{ -}\mathsf{TC}^{{0}}.$$
It is unknown if $$\mathsf{PP}\not\subseteq\mathsf{LOGSPACEuniform}\mbox{ -}\mathsf{TC}^{{0}}$$ holds (What is largest class of functions $C$ such that we know $\#P$ in not contained in $C$-generated $TC^0$?).
Possibility of 2. is the deeper collapse.
Update The comments below indicate following collapse is not ruled out: $$\mathsf{CH}\subseteq\mathsf{LOGSPACEuniform}\mbox{-}\mathsf{TC}^0\subseteq\mathsf{LOGSPACEuniform}\mbox{-}\mathsf{ACC}^0.$$