  Goyeneche et al.\ [Phys.\ Rev.\ A \textbf{97}, 062326 (2018)] introduced
several classes of quantum combinatorial designs, namely quantum Latin squares,
quantum Latin cubes, and the notion of orthogonality on them. They also showed
that mutually orthogonal quantum Latin arrangements can be entangled in the
same way in which quantum states are entangled. Moreover, they established a
relationship between quantum combinatorial designs and a remarkable class of
entangled states called $k$-uniform states, i.e., multipartite pure states such
that every reduction to $k$ parties is maximally mixed. In this article, we put
forward the notions of incomplete quantum Latin squares and orthogonality on
them and present construction methods for mutually orthogonal quantum Latin
squares and mutually orthogonal quantum Latin cubes. Furthermore, we introduce
the notions of generalized mutually orthogonal quantum Latin squares and
generalized mutually orthogonal quantum Latin cubes, which are equivalent to
quantum orthogonal arrays of size $d^2$ and $d^3$, respectively, and thus
naturally provide $2$- and $3$-uniform states.
