  The interplay between single-particle interference and quantum
indistinguishability leads to signature correlations in many-body scattering.
We uncover these with a semiclassical calculation of the transmission
probabilities through mesoscopic cavities for systems of non-interacting
particles. For chaotic cavities we provide the universal form of the first two
moments of the transmission probabilities over ensembles of random unitary
matrices, including weak localization and dephasing effects. If the incoming
many-body state consists of two macroscopically occupied wavepackets, their
time delay drives a quantum-classical transition along a boundary determined by
the bosonic birthday paradox. Mesoscopic chaotic scattering of Bose-Einstein
condensates is then a realistic candidate to build a boson sampler and to
observe the macroscopic Hong-Ou-Mandel effect.
