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This paper considers the joint device activity detection and channel estimation problem in a massive Internet of Things (IoT) connectivity system, where a large number of IoT devices exist but merely a random subset of them become active for short-packet transmission in each coherence block. In particular, we propose to leverage the temporal correlation in device activity, e.g., a device active in the previous coherence block is more likely to be still active in the current coherence block, to improve the detection and estimation performance. However, it is challenging to utilize this temporal correlation as side information (SI), which relies on the knowledge about the exact statistical relation between the estimated activity pattern for the previous coherence block (which may be imperfect with unknown error) and the true activity pattern in the current coherence block. To tackle this challenge, we establish a novel SI-aided multiple measurement vector approximate message passing (MMV-AMP) framework. Specifically, thanks to the state evolution of the MMV-AMP algorithm, the correlation between the activity pattern estimated by the MMV-AMP algorithm in the previous coherence block and the real activity pattern in the current coherence block is quantified explicitly. Based on the well-defined temporal correlation, we further manage to embed this useful SI into the denoiser design under the MMV-AMP framework. Specifically, the SI-based soft-thresholding denoisers with binary thresholds and the SI-based minimum mean-squared error (MMSE) denoisers are characterized for the cases without and with the knowledge of the channel distribution, respectively. Numerical results are given to show the significant gain in device activity detection and channel estimation performance brought by our proposed SI-aided MMV-AMP framework.