There are several well-known $\mathsf{AC^0}$ circuit size lower-bound results based on random restrictions and the Switching Lemma.Can we develop a Switching Lemma result to prove a size lower-bound for $\mathsf{TC^0}$ circuits (similar to the lower-bound proofs for $\mathsf{AC^0}$)? Or is there any inherent obstacle to using this approach for proving $\mathsf{TC^0}$ lower-bounds? Do barrier results like Natural Proofs say anything regarding using Switching Lemma like techniques to prove $\mathsf{TC^0}$ lower-bounds?
Is it possible to use random restrictions to obtain a lower-bound for $\mathsf{TC^0}$?
cc.complexity theory;circuit complexity;lower bounds;boolean functions