= Square-power rigidity of a finite nonzero spectrum
Let $S$ be a finite multiset of nonzero complex numbers such that taking the $m^2$th powers permutes it for every positive integer $m$. Taking $m=2$ and iterating the resulting permutation shows that each $z\in S$ obeys $z^{4^r}=z$ for some $r\geq1$, hence is a <root of unity>. Let $d$ be a common multiple of their orders. The $d^2$th powers are all one, and the assumed permutation then forces every original element to be one. For a <unitary matrix>, the <spectral theorem for normal operators> further forces the matrix itself to be the identity.
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