Alzheimer's disease (AD) is a neurodegenerative disorder with progressive impairment of memory and cognitive functions. Sparse coding (SC) has been demonstrated to be an efficient and effective method for AD diagnosis and prognosis. However, previous SC methods usually focus on the baseline data while ignoring the consistent longitudinal features with strong sparsity pattern along the disease progression. Additionally, SC methods extract sparse features from image patches separately rather than learn with the dictionary atoms across the entire subject. To address these two concerns and comprehensively capture temporal-subject sparse features towards earlier and better discriminability of AD, we propose a novel supervised SC network termed Temporally Adaptive-Dynamic Sparse Network (TADsNet) to uncover the sequential correlation and native subject-level codes from the longitudinal brain images. Our work adaptively updates the sparse codes to impose the temporal regularized correlation and dynamically mine the dictionary atoms to make use of entire subject-level features. Experimental results on ADNI-I cohort validate the superiority of our approach.