  Suppose we have $N$ quantum systems in unknown states $\lvert\psi_i \rangle
$, but know the value of some pairwise overlaps $\left| \langle \psi_k \lvert
\psi_l \rangle \right|^2$. What can we say about the values of the unknown
overlaps? We provide a complete answer to this problem for three pure states
and two given overlaps, and a way to obtain bounds for the general case. We
discuss how the answer contrasts from that of a classical model featuring only
coherence-free, diagonal states, and describe three applications:
basis-independent coherence witnesses, dimension witnesses, and
characterisation of multi-photon indistinguishability.
