We provide new query complexity separations against sensitivity for total Boolean functions: a power 3 separation between deterministic (and even randomized or quantum) query complexity and sensitivity, and a power 2.22 separation between certificate complexity and sensitivity. We get these separations by using a new connection between sensitivity and a seemingly unrelated measure called one-sided unambiguous certificate complexity. We also show that one-sided unambiguous certificate complexity is lower-bounded by fractional block sensitivity, which means we cannot use these techniques to get a super-quadratic separation between block sensitivity and sensitivity.
Along the way, we give a power 1.22 separation between certificate complexity and one-sided unambiguous certificate complexity, improving the power 1.128 separation due to Göös (FOCS 2015). As a consequence, we obtain an improved lower-bound on the co-nondeterministic communication complexity of the Clique vs. Independent Set problem.