Mild cognitive impairment (MCI) patients are at a high risk of turning into Alzheimer's disease (AD) within years. But it is known that not all MCI patients will progress to AD. Therefore, it is of great interest to accurately diagnose whether a MCI patient will convert to AD (namely MCI converter; MCI-C) or not (namely MCI non-converter; MCI-NC), for early diagnosis and proper treatment. In this paper, we propose a multi-task sparse representation classifier to discriminate between MCI-C and MCI-NC utilizing longitudinal neuroimaging data. Unlike the previous methods that explicitly combined the longitudinal information in a feature domain, thus requiring the same number of measurements in time, the proposed method is not limited to the availability of the data. Specifically, by means of multi-task learning, we impose a group constraint that the same training samples, ideally belonging to the same class, are used to represent the longitudinal feature vectors across time points. Then we utilize a sparse representation classifier for label decision. From a machine learning perspective, the proposed method can be considered as the combination of the generative and discriminative methods, which are known to be effective in classification enhancement. In our experiments on magnetic resonance brain images of 349 MCI subjects (164 MCI-C and 185 MCI-NC) from ADNI database, we demonstrate the validity of the proposed method, which also outperforms the competing methods.