The parallel dynamics of the fully connected Blume-Emery-Griffiths neural network model is studied at zero temperature using a probabilistic approach. A recursive scheme is found determining the complete time evolution of the order parameters, taking into account all feedback correlations. It is based upon the evolution of the distribution of the local field, the structure of which is determined in detail. As an illustrative example explicit analytic formula are given for the first few time steps of the dynamics. Furthermore, equilibrium fixed-point equations are derived and compared with the thermodynamic approach. The analytic results find excellent confirmation in extensive numerical simulations.Recently, an optimal Hamiltonian has been derived in the statistical mechanics approach to Qstate neural networks starting from the concept of mutual information [1], [2]. Optimal means that the best retrieval properties are guaranteed including, e.g., the largest retrieval overlap, loading capacity, basin of attraction, convergence time. For Q = 3 this Hamiltonian resembles the classical Blume-Emery-Griffiths (BEG) Hamiltonian in the sense that it contains both a bilinear and biquadratic term in the spins [3]. For a fully connected architecture it has been shown, using a thermodynamic replica approach that the maximal loading capacity for the BEG network is indeed bigger than the one for other three-state networks existing in the literature (Ising, Potts...) [2].