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
(UCmin). We also show that UCmin is lower-bounded by fractional block
sensitivity, which means we cannot use these techniques to get a
super-quadratic separation between bs(f) and s(f). We also provide a
quadratic separation between the tree-sensitivity and decision tree complexity
of Boolean functions, disproving a conjecture of Gopalan, Servedio, Tal, and
Wigderson (CCC 2016).
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 G\"o\"os (FOCS 2015). As a consequence, we
obtain an improved Ω(log1.22n) lower-bound on the
co-nondeterministic communication complexity of the Clique vs. Independent Set
problem.Comment: 25 pages. This version expands the results and adds Pooya Hatami and
Avishay Tal as author