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Baryon number conservation is not guaranteed by any fundamental symmetry within the standard model, and therefore has been a subject of experimental and theoretical scrutiny for decades. So far, no evidence for baryon number violation has been observed. Large underground detectors have long been used for both neutrino detection and searches for baryon number violating processes. The next generation of large neutrino detectors will seek to improve upon the limits set by past and current experiments and will cover a range of lifetimes predicted by several Grand Unified Theories. In this White Paper, we summarize theoretical motivations and experimental aspects of searches for baryon number violation in neutrino experiments.

The next core-collapse supernova in the Milky Way or its satellites will represent a once-in-a-generation opportunity to obtain detailed information about the explosion of a star and provide significant scientific insight for a variety of fields because of the extreme conditions found within. Supernovae in our galaxy are not only rare on a human timescale but also happen at unscheduled times, so it is crucial to be ready and use all available instruments to capture all possible information from the event. The first indication of a potential stellar explosion will be the arrival of a bright burst of neutrinos. Its observation by multiple detectors worldwide can provide an early warning for the subsequent electromagnetic fireworks, as well as signal to other detectors with significant backgrounds so they can store their recent data. The supernova early warning system (SNEWS) has been operating as a simple coincidence between neutrino experiments in automated mode since 2005. In the current era of multi-messenger astronomy there are new opportunities for SNEWS to optimize sensitivity to science from the next galactic supernova beyond the simple early alert. This document is the product of a workshop in June 2019 towards design of SNEWS 2.0, an upgraded SNEWS with enhanced capabilities exploiting the unique advantages of prompt neutrino detection to maximize the science gained from such a valuable event.