  We introduce several classes of quantum combinatorial designs, namely quantum
Latin squares, cubes, hypercubes and a notion of orthogonality between them. A
further introduced notion, quantum orthogonal arrays, generalizes all previous
classes of designs. We show that mutually orthogonal quantum Latin arrangements
can be entangled in the same way than quantum states are entangled.
Furthermore, we show that such designs naturally define a remarkable class of
genuinely multipartite highly entangled states called $k$-uniform, i.e.
multipartite pure states such that every reduction to $k$ parties is maximally
mixed. We derive infinitely many classes of mutually orthogonal quantum Latin
arrangements and quantum orthogonal arrays having an arbitrary large number of
columns. The corresponding multipartite $k$-uniform states exhibit a high
persistency of entanglement, which makes them ideal candidates to develop
multipartite quantum information protocols.
