  We show that (1) the violation of the Ekert 91 inequality is a sufficient
condition for certification of the Kochen-Specker (KS) theorem, and (2) the
violation of the Bennett-Brassard-Mermin 92 (BBM) inequality is, also, a
sufficient condition for certification of the KS theorem. Therefore the success
in each QKD protocol reveals the nonclassical feature of quantum theory, in the
sense that the KS realism is violated. Further, it turned out that the Ekert
inequality and the BBM inequality are depictured by distillable entanglement
witness inequalities. Here, we connect the success in these two key
distribution processes into the no-hidden-variables theorem and into witness on
distillable entanglement. We also discuss the explicit difference between the
KS realism and Bell's local realism in the Hilbert space formalism of quantum
theory.
