We develop and demonstrate a trainable temporal post-processor (TPP), harnessing a simple but versatile machine learning algorithm to provide optimal processing of quantum measurement data subject to arbitrary noise processes, for the readout of an arbitrary number of quantum states. We demonstrate the TPP on the essential task of qubit state readout, which has historically relied on temporal processing via matched filters in spite of their applicability only for specific noise conditions. Our results show that the TPP can reliably outperform standard filtering approaches under complex readout conditions, such as high power readout. Using simulations of quantum measurement noise sources, we show that this advantage relies on the TPP's ability to learn optimal linear filters that account for general quantum noise correlations in data, such as those due to quantum jumps, or correlated noise added by a phase-preserving quantum amplifier. Furthermore, for signals subject to Gaussian white noise processes, the TPP provides a linearly-scaling semi-analytic generalization of matched filtering to an arbitrary number of states. The TPP can be efficiently, autonomously, and reliably trained on measurement data, and requires only linear operations, making it ideal for FPGA implementations in cQED for real-time processing of measurement data from general quantum systems.