  In order to use quantum devices for computations, it is necessary to
understand the intricacies of the theoretical description. To this end, we
provide several novel constructions useful for the comprehension of quantum
mechanics from the perspective of mappings and designs. The unistochasticity
problem, which relates the classical and the quantum domain, is solved in
specific cases, e.g. for all matrices of dimension 4. Furthermore, we provide
an explicit formula for entangling power in the multipartite case. Most
importantly, the thesis presents and elaborates on the path that lead to the
recent construction of absolutely maximally entangled state of four subsystems
six levels each. Finally, we study cardinality as a measure of "quantumness" of
quantum Latin squares and quantum Sudoku (SudoQ). Arrays of the highest
cardinality yield families of quantum measurements of special properties. A
connection between SudoQ designs and mutually unbiased bases is demonstrated.
