  We prove lower bounds on complexity measures, such as the approximate degree
of a Boolean function and the approximate rank of a Boolean matrix, using
quantum arguments. We prove these lower bounds using a quantum query algorithm
for the combinatorial group testing problem.
  We show that for any function f, the approximate degree of computing the OR
of n copies of f is Omega(sqrt{n}) times the approximate degree of f, which is
optimal. No such general result was known prior to our work, and even the lower
bound for the OR of ANDs function was only resolved in 2013.
  We then prove an analogous result in communication complexity, showing that
the logarithm of the approximate rank (or more precisely, the approximate
gamma_2 norm) of F: X x Y -> {0,1} grows by a factor of Omega~(sqrt{n}) when we
take the OR of n copies of F, which is also essentially optimal. As a
corollary, we give a new proof of Razborov's celebrated Omega(sqrt{n}) lower
bound on the quantum communication complexity of the disjointness problem.
  Finally, we generalize both these results from composition with the OR
function to composition with arbitrary symmetric functions, yielding nearly
optimal lower bounds in this setting as well.
