  In this work, two novel classes of structure-preserving spectral Galerkin
methods are proposed which based on the Crank-Nicolson scheme and the
exponential scalar auxiliary variable method respectively, for solving the
coupled fractional nonlinear Klein-Gordon-Schr\"odinger equation. The paper
focuses on the theoretical analyses and computational efficiency of the
proposed schemes, the Crank-Nicoloson scheme is proved to be unconditionally
convergent and has the maximum-norm boundness of numerical solutions. The
exponential scalar auxiliary variable scheme is linearly implicit and
decoupled, but lack of the maximum-norm boundness, also, the energy structure
has been modified. Subsequently, the efficient implementations of the proposed
schemes are introduced in detail. Both the theoretical analyses and the
numerical comparisons show that the proposed spectral Galerkin methods have
high efficiency in long-time computations.
