Separations in query complexity for total search problems

We study the query complexity analogue of the class TFNP of total search problems. We give a way to convert partial functions to total search problems under certain settings; we also give a way to convert search problems back into partial functions. As an application, we give new separations for degree-like measures. We give an exponential separation between quantum query complexity and approximate degree for a total search problem. We also give an exponential separation between approximate degree and the positive quantum adversary for a total search problem. We then strengthen the former separation to upper bound a larger measure: the two-sided approximate non-negative degree, also called the conical junta degree. This measure is often larger than quantum query complexity and even a separation from randomized query complexity was not known. We extend our results to communication complexity, and obtain an exponential separation between quantum information complexity and the relaxed partition bound for a total search problem. Even a weaker separation between randomized communication complexity and the relaxed partition bound was not known for total search problems (or even for partial functions). Most of our separations for total search problems can be converted to separations for partial functions. Using this, we reprove the recent exponential separation between quantum query complexity and approximate degree for a partial function by Ambainis and Belovs (2023), among other new results.