Local realism cannot impose commutative structure into noncommuting quantum observables

We formally prove the ultimate statement: Local realistic theories for quantum measurements could not impose the commutative algebraic structure into the set of all quantum observables even if all quantum statistics (including every quantum state and every observable) could be reproduced by local realistic theories. Thus, it turns out that the result, using a pair of projectors, by Malley et al. [Phys. Lett. A 347, 51 (2005)] is applicable only to the Kochen-Specker (KS) theorem. Moreover, we also point out that an inefficient detector cannot be described by a projector. Hence, the proposal of empirical tests of Malley [Phys. Rev. A 58, 812 (1998)] for the KS theorem cannot rule out quantum efficiency issues and it cannot apply to realizable experiments. We also discuss nonlocality in the hidden-variable theory of the KS type in the Hilbert space formalism of quantum theory.