Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 67 / 2 / b

For the three-bit majority function, the pair products count how many pairs of input bits are both one. Their sum is zero at Hamming weights zero and one, one at weight two, and three at weight three. Subtracting twice the triple product corrects the last value. HenceThis is a multilinear polynomial of degree three, and its top coefficient is nonzero. Uniqueness of the Boolean multilinear polynomial excludes a degree-two alternative. The polynomial method for quantum query lower bounds therefore yields $T\ge 3/2$, and the integer number of queries gives