How do Latin squares relate to quantum cryptography?
There is a conjecture that a full set of mutually orthogonal Latin squares' existence is linked to the existence of MUBs. I am unsure if there is support for this beyond experimental evidence.

Mutually Unbiased Bases (MUBs) introduced by Schwinger are used in quantum measurements and also quantum cryptography. MUBs are $d+1$ Orthonormal bases of $\mathbb{C}^d$ where the inner product of any pair of vectors between two distinct bases is $1/\sqrt{d}$ in magnitude. 

For example, constructing
 a complete set of mutually unbiased bases in the $\mathbb{C}^d$ with $d$ composite and not a prime power has not been achieved yet, not even for $d=6.$

See the PhD thesis of Joanne Hall here