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We prove new lower bounds for bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by R. Raz (1995) to the quantum case. Applying this method we give an exponential separation between bounded error quantum communication complexity and nondeterministic quantum communication complexity. We develop several other Fourier based lower bound methods, notably showing that √(s~(f)/log n) n, for the average sensitivity s~(f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f (x∧y⊕yz), where x is a Boolean word held by Alice and y, z are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, only the previously applied general lower bound method based on discrepancy yields bounds that are O(log n).