Lensless imagers based on diffusers or encoding masks enable high-dimensional imaging from a single shot measurement and have been applied in various applications. However, to further extract image information such as edge detection, conventional post-processing filtering operations are needed after the reconstruction of the original object images in the diffuser imaging systems. Here, we present the concept of a temporal compressive edge detection method based on a lensless diffuser camera, which can directly recover a time sequence of edge images of a moving object from a single-shot measurement, without further post-processing steps. Our approach provides higher image quality during edge detection, compared with the conventional post-processing method. We demonstrate the effectiveness of this approach by both numerical simulation and experiments. The proof-of-concept approach can be further developed with other image post-process operations or versatile computer vision assignments toward task-oriented intelligent lensless imaging systems.