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We construct a total Boolean function $f$ satisfying$R(f)=\tilde{\Omega}(Q(f)^{5/2})$, refuting the long-standing conjecture that$R(f)=O(Q(f)^2)$ for all total Boolean functions. Assuming a conjecture ofAaronson and Ambainis about optimal quantum speedups for partial functions, weimprove this to $R(f)=\tilde{\Omega}(Q(f)^3)$. Our construction is motivated bythe G\"o\"os-Pitassi-Watson function but does not use it.

A Super-Grover Separation Between Randomized and Quantum Query Complexities