In this paper, we are concerned with the existence of sets of mutually quasi-orthogonal Latin squares (MQOLS). We establish a correspondence between equidistant permutation arrays and MQOLS, which has facilitated a computer search to identify all sets of MQOLS of order \(\leq 6\). In particular, we report that the maximum number of Latin squares of order 6 in a mutually quasi-orthogonal set is 3, and give an example of such a set. We also report on a non-exhaustive computer search for sets of 3 MQOLS of order 10, which, whilst not identifying such a set, has led to the identification of all the resolutions of each \((10, 3, 2)\)-balanced incomplete block design. Improvements are given on the existence results for MQOLS based on groups, and a new construction is given for sets of MQOLS based on groups from sets of mutually orthogonal Latin squares based on groups. We show that this construction yields sets of \(2^n – 1\) MQOLS of order \(2^n\), based on two infinite classes of groups. Finally, we give a new construction for difference matrices from mutually quasi-orthogonal quasi-orthomorphisms, and use this to construct a \((2^n, 2^n; 2)\)-difference matrix over \({C}_2^{n-2} \times {C}_4\).