Timely perception of distribution system states is critical for the control and operation of power grids. Recently, it has been seriously challenged by the dramatic voltage fluctuations induced by high renewables. To address this issue, an Unrolled Spatiotemporal Graph Convolutional Network (USGCN) is proposed for distribution system state estimation (DSSE) and forecasting with augmented consideration of the underlying complex spatiotemporal correlations of renewable energy sources (RES). Specifically, the interconnection among individual spatial graphs of adjacent time steps will lead to an unrolled spatiotemporal graph and benefit the synchronous capture of spatial and temporal correlations to achieve enhanced accuracy. On top of this, the node-embedding technique is employed in the unrolled spatiotemporal convolutional layer to reveal the hidden nonlinear spatiotemporal correlations of RES outputs without relying on full prior knowledge. Moreover, the proposed USGCN stacks the unrolled spatiotemporal convolutional layers, leading to the perception of longtime correlations to obtain effective ahead-of-time state forecasting results robustly. The simulation results have been provided to verify the accuracy and efficiency of the proposed model in 118-node and 1746-node distribution systems.