  Hidden-variables models are critically reassessed. It is first examined if
the quantum discord is classically described by the hidden-variable model of
Bell in the Hilbert space with $d=2$. The criterion of vanishing quantum
discord is related to the notion of reduction and, surprisingly, the
hidden-variable model in $d=2$, which has been believed to be consistent so
far, is in fact inconsistent and excluded by the analysis of conditional
measurement and reduction. The description of the full contents of quantum
discord by the deterministic hidden-variables models is not possible. We also
re-examine CHSH inequality. It is shown that the well-known prediction of CHSH
inequality $|B|\leq 2$ for the CHSH operator $B$ introduced by Cirel'son is not
unique. This non-uniqueness arises from the failure of linearity condition in
the non-contextual hidden-variables model in $d=4$ used by Bell and CHSH, in
agreement with Gleason's theorem which excludes $d=4$ non-contextual
hidden-variables models. If one imposes the linearity condition, their model is
converted to a factored product of two $d=2$ models which describes quantum
mechanical separable states. The CHSH inequality thus does not test the
hidden-variables model in $d=4$. This observation is consistent with an
application of the CHSH inequality to quantum cryptography by Ekert, which is
based on mixed separable states without referring to dispersion-free
representations. As for hidden-variables models, there exist no viable local
non-contextual models in any dimensions.
