Separating TC^0 from NC^1

Prove or disprove that the circuit complexity classes TC^0 and NC^1 are unequal, i.e., establish whether TC^0 ≠ NC^1.

Background

The paper’s expressivity results for transformers and SSMs are interpreted within the standard circuit complexity hierarchy. Their inexpressivity claims for tasks like permutation composition rely on the widely held (but unproven) conjecture that TC^0 is strictly contained in NC^1.

Resolving this conjecture would have fundamental implications for parallel vs. inherently sequential computation and would solidify class-separation-based arguments used to analyze neural model expressivity.