The performance of sparse signal recovery (SSR) can be enhanced by exploiting rich temporal correlation in the multiple snapshots of signal of interest. However, existing methods need to transform the temporally correlated multiple measurements SSR problem into its vectorization form, imposing huge computational cost for algorithmic realization. To overcome this drawback, we propose a novel formulation to model the temporal correlation so that variational Bayesian inference (VBI) can be applied to simplify the inference and a novel uncoupling trick is also proposed to reduce the computation. Theoretical and simulation results indicate that our method can bring a considerable computational complexity reduction and achieve a performance improvement for the temporally correlated SSR problem compared to the state of arts time-varying sparse Bayesian learning (TSBL) method.