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We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity and approximate(More)
Large databases with uncertain information are becoming more common in many applications including data integration, location tracking, and Web search. In these applications , ranking records with uncertain attributes introduces new problems that are fundamentally different from conventional ranking. Specifically, uncertainty in records' scores induces a(More)
While exponential separations are known between quantum and randomized communication complexity for partial functions (Raz, STOC 1999), the best known separation between these measures for a total function is quadratic, witnessed by the disjointness function. We give the first super-quadratic separation between quantum and randomized communication(More)
We construct a total Boolean function f satisfying R(f) = ˜ Ω(Q(f) 5/2), refuting the long-standing conjecture that R(f) = O(Q(f) 2) for all total Boolean functions. Assuming a conjecture of Aaronson and Ambainis about optimal quantum speedups for partial functions, we improve this to R(f) = ˜ Ω(Q(f) 3). Our construction is motivated by the(More)
We study the composition question for bounded-error randomized query complexity: Is R(f • g) = Ω(R(f)R(g)) for all Boolean functions f and g? We show that inserting a simple Boolean function h, whose query complexity is only Θ(log R(g)), in between f and g allows us to prove R(f • h • g) = Ω(R(f)R(h)R(g)). We prove this using a new lower bound measure for(More)
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 2.1 separation between certificate complexity and sensitivity. We get these separations by using a new connection between sensitivity and a(More)
Two separate results related to decision-tree complexity are presented. The first uses a topological approach to generalize some theorems about the evasiveness of monotone boolean functions to other classes of functions. The second bounds the gap between the deterministic decision-tree complexity of functions on the permutation group S n and their(More)