Building on previous works, this paper brings novel results about the quaternion state-space mathematical modeling, under he assumptions of rate gyro measurements and of vector observations of the attitude. The spectral properties of the Lyapunov differential equation for the quaternion second-order moment are investigated and expressions for its solution are developed. The properties of a four-dimensional quaternion measurement matrix are analyzed. Three different approaches are suggested in order to reduce, without loosing information, the measurement equation down to a two-dimensional equation. The insight gained in this analysis leads to the development of a deterministic quaternion estimator from a batch of two vector observations. The reduced quaternion measurement model yields computationally more efficient quaternion Kalman filters. The relationship between the proposed filters and the celebrated q-method is discussed.