A simple nonlocal theory is put into Hamiltonian form and quantized by using the modern version of Ostrogradski approach.

In two recent papers (Phys. Rev. A90 (2014), 062121 and Phys. Rev. (2015), 052110) an interesting method of analyzing the violation of Bell inequalities has been proposed which is based on the theory of finite group representations. We apply here this method to more complicated example of S 4 symmetry. We show how the Bell inequality can be related to the symmetries of regular tetrahedron. By choosing the orbits of threedimensional representation of S 4 determined by the geometry of tetrahedron we find that the Bell inequality under consideration is violated in quantum theory. The corresponding nonlocal game is analyzed. A91

We analyze certain aspects of group theoretical approach to Bell inequalities proposed by Güney and Hillery. The general procedure for constructing the relevant group orbits is described. Using Hall theorem we determine the form of Bell inequality in the single orbit case. It is shown that in this case the Bell inequality is not violated for maximally entangled state generating trivial subrepresentation if the representation under consideration is real.

The recently proposed (Güney and Hillery in Phys Rev A 90:062121, 2014; Phys Rev A 91:052110, 2015) group theoretical approach to the problem of violating the Bell inequalities is applied to S 4 group. The Bell inequalities based on the choice of three orbits in the representation space corresponding to standard representation of S 4 are derived and their violation is described. The corresponding nonlocal games are analyzed.

The two-players N strategies games quantized according to the Eisert-Lewenstein-Wilkens scheme (Phys. Rev. Lett. 83 (1999), 3077) are considered. Group theoretical methods are applied to the problem of finding a general form of gate operators (entanglers) under the assumption that the set of classical pure strategies is contained in the set of pure quantum ones.The role of the stability group of the initial state of the game is stressed.As an example, it is shown that the maximally entangled games do not admit nontrivial pure Nash strategies. The general arguments are supported by explicit computations performed in the three strategies case.