  It is known that the local Yang--Baxter equation is a generator of potential
solutions to Zamolodchikov's tetrahedron equation. In this paper, we show under
which additional conditions the solutions to the local Yang--Baxter equation
are tetrahedron maps, namely solutions to the set-theoretical tetrahedron
equation. This is exceptionally useful when one wants to prove that
noncommutative maps satisfy the Zamolodchikov's tetrahedron equation. We
construct new noncommutative maps and we prove that they possess the
tetrahedron property. Moreover, by employing Darboux transformations with
noncommutative variables, we derive noncommutative tetrahedron maps. In
particular, we derive a noncommutative nonlinear Schr\"odinger type of
tetrahedron map which can be restricted to a noncommutative version of
Sergeev's map on invariant leaves. We prove that these maps are tetrahedron
maps.
