  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 polynomial degree, showing severe
limitations on the power of the polynomial method. Finally, we exhibit a total
function with a quadratic gap between quantum query complexity and certificate
complexity, which is optimal (up to log factors). These separations are shown
using a new, general technique that we call the cheat sheet technique. The
technique is based on a generic transformation that converts any (possibly
partial) function into a new total function with desirable properties for
showing separations. The framework also allows many known separations,
including some recent breakthrough results of Ambainis et al., to be shown in a
unified manner.
