Can worst-case trace reconstruction be done with polynomially many traces?

Theorem 1.1: for fixed $s\in(0,1]$ and $0<c<1/(4\log 2)$, $T_{q,s}(n)\ge n^{c\log(q^3\log n)}$ eventually whenever $q^3\log n\to\infty$; so $T_q(n)=n^{\Omega(\log\log n)}$ at every fixed $q$, a negative answer to the polynomial question. Theorem 1.2: the minimum total variation between one-trace laws of distinct length-$n$ words is $o(n^{-A})$ for all $A$. The companions give a matching-type upper side: quasipolynomial samples and bit complexity with a uniform decoder for known rational retention, and polynomial bounds when $q\le n^{-\varepsilon}$. A gap remains between $\log T$ of order $\log n\log\log n$ (lower) and a power of $\log n$ (upper); the exact complexity is not determined.

Theorem 1.1: for fixed $s\in(0,1]$ and $0<c<1/(4\log 2)$, $T_{q,s}(n)\ge n^{c\log(q^3\log n)}$ eventually whenever $q^3\log n\to\infty$; so $T_q(n)=n^{\Omega(\log\log n)}$ at every fixed $q$, a negative answer to the polynomial question. Theorem 1.2: the minimum total variation between one-trace laws of distinct length-$n$ words is $o(n^{-A})$ for all $A$. The companions give a matching-type upper side: quasipolynomial samples and bit complexity with a uniform decoder for known rational retention, and polynomial bounds when $q\le n^{-\varepsilon}$. A gap remains between $\log T$ of order $\log n\log\log n$ (lower) and a power of $\log n$ (upper); the exact complexity is not determined.