Extension and optimality of polynomial indistinguishability

Tighten the polynomial indistinguishability result for Kac matrices by extending indistinguishability against degree-k polynomials from the regime k approximately at most the square root of n to k approximately n, and determine whether a walk length of roughly n k squared is necessary for indistinguishability against degree-k polynomials.

Background

The main theorem establishes that degree-k polynomials cannot distinguish a sufficiently long Kac walk from Haar measure, with the relevant walk length scaling roughly as n k squared up to logarithmic factors. The authors note that the theorem ceases to be informative when k is about the square root of n and explicitly leave unresolved both the possibility of extending the result to k about n and the necessity of the n k squared running-time dependence.

References

Can we tighten the polynomial indistinguishability result? First, recall that our Theorem~\ref{MainThm} stops being interesting for $k \approx \sqrt{n}$: can we extend this all the way to $k \approx n$? Secondly, to be indistinguishable against degree-$k$ polynomials, we need the walk to run for roughly $nk2$ steps: is that necessary?

— On the Pseudo-Mixing of Kac's Walk  (2608.17374 - Pillai et al., 18 Aug 2026) in Section “Open Questions,” Subsection “Immediate Open Questions”

Finally, a very natural question is whether low-degree polynomials of the {\em spectrum} of a Kac matrix can be a good distinguisher. It seems plausible that the answer is no: the basic intuition here is just that the spectral measure is a nice'' function of only npretty independent'' numbers, so should mix in $n \mathsf{poly}(\log n)$ steps. However, making this argument formally appears more difficult, since the spectrum is not quite a Markov chain (in contrast to the columns that we analyze here, which are).

— On the Pseudo-Mixing of Kac's Walk  (2608.17374 - Pillai et al., 18 Aug 2026) in Section “Open Questions,” Subsection “Immediate Open Questions”