We compare the performance of the Kalman filter (KF)-based and the minimum variance (MV) control algorithms for a zonal adaptive optics with a phase temporal prediction step included for effective compensation of the errors attributable to latencies in the system. The main goal is to evaluate the performance achievable by the computationally more expensive KF approach, which explicitly accounts for the atmospheric turbulence temporal behavior through a first-order autoregressive evolution model, and the simpler MV algorithm, with and without temporal prediction. For a representative example, the Gemini-South 8 m telescope multiconjugate adaptive optics system performance of the KF and the MV controllers has been compared with respect to their turbulence compensation capability. We show that the KF algorithm, as expected, shows superior performance to that of the MV algorithm, especially for extremely low sampling rates and large control latencies. We also show that for moderate control latencies the MV algorithm with a temporal prediction step added to it approaches the performance of the KF technique.