Restoring discrete quantized signal I have a discrete signal that can only take two values - 0 and 1. Signal is band limited with limiting frequency 1MHz. I sample this signal with frequency 1KHz. Question - can I do better than nearest-neighbor reconstruction or zero-order hold reconstruction? Bonus question: what type of information can be used to increase reconstruction accuracy (other than increasing sampling frequency, obviously)?
A qualitative signal recovery is possible provided that its parametric model is known. In this case, the quality of signal recovery will be determined by the accuracy of restoring its parameters and that part of the original signal that is not taken into account in the model.

In this case, the reconstruction methods can be different: spectral analysis, the method of least entropy, polynomial regression analysis, and others.
if the signal is bandlimited to 1 MHz, there is no way that sampling at 1 kHz suffices to get any information.  the signal would have to be bandlimited to below 500 Hz in order for sampling with 1000 samples per second to suffice.

this is standard Nyquist-Shannon sampling theorem.