In the paper, we consider the coupled nonlinear Schrodinger equation with high degree polynomials in the energy functional that cannot be handled by using the newly proposed quadratic auxiliary variable method. Therefore, we develop the multiple quadratic auxiliary variable approach to deal with coupled systems and construct high-accuracy structure-preserving schemes for the  equation. To fix the idea, we first apply the multiple quadratic auxiliary variable approach to the equation and obtain an equivalent system that possesses the original energy and mass. Then, a family of high-accuracy structure-preserving schemes that can conserve the mass and energy is derived by applying the Gauss collocation method and sine pseudo-spectral method to approximate the system in time and space. The given schemes have high-accuracy in time and can both inherit the mass and Hamiltonian energy of the system. Ample numerical results are given to confirm the accuracy and conservation of the developed schemes at last.