Open-problem statements, with their sources attached.

Benchmark open subproblem: existence of a SIC-POVM in dimension $56$.

Benchmark open subproblem: existence of a SIC-POVM in dimension $58$.

Benchmark open subproblem: existence of a SIC-POVM in dimension $59$.

Benchmark open subproblem: existence of a SIC-POVM in dimension $60$.

Benchmark open subproblem: existence of a SIC-POVM in dimension $64$.

Benchmark open subproblem: existence of a SIC-POVM in dimension $68$.

Benchmark open subproblem: existence of a SIC-POVM in dimension $69$.

Benchmark open subproblem: existence of a SIC-POVM in dimension $70$.

Benchmark open subproblem: existence of a SIC-POVM in dimension $71$.

Benchmark open subproblem: existence of a SIC-POVM in dimension $72$.

Benchmark open subproblem: existence of a SIC-POVM in dimension $75$.

Do SIC-POVMs exist in every finite dimension?

Does the determinant of the sum $A + B$ of two $n \times n$ normal complex matrices $A$ and $B$ always lie in the convex hull of the $n!$ points $\prod_i (\lambda(A)_i + \lambda(B)_{\sigma(i)})$? Here the numbers $\lambda(A)_i$ and $\lambda(B)_i$ are the eigenvalues of $A$ and $B$, and $\sigma$ is an element of the symmetric group $S_n$.

There exists a Hadamard matrix for all $n = 4k$.

The smallest order for which no Hadamard matrix is presently known is $668 = 4 * 167$.