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?

Special case in dimension $6$: determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^6$.

Special case in dimension $10$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{10}$.

Special case in dimension $12$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{12}$.

Special case in dimension $14$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{14}$.

Special case in dimension $15$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{15}$.

Open Quantum Problem 13: determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^d$ for $d \ge 2$.

Open benchmark statement: does an $\mathrm{AME}(7,6)$ state exist?

Open benchmark statement: does an $\mathrm{AME}(7,10)$ state exist?