  We test the principle of majorization [J. I. Latorre and M. A.
Martin-Delgado, Phys. Rev. A 66, 022305 (2002)] in random circuits. Three
classes of circuits were considered: (i) universal, (ii) classically
simulatable, and (iii) neither universal nor classically simulatable. The
studied families are: {CNOT, H, T}, {CNOT, H, NOT}, {CNOT, H, S} (Clifford),
matchgates, and IQP (instantaneous quantum polynomial-time). We verified that
all the families of circuits satisfy on average the principle of decreasing
majorization. In most cases the asymptotic state (number of gates going to
infinity) behaves like a random vector. However, clear differences appear in
the fluctuations of the Lorenz curves associated to asymptotic states. The
fluctuations of the Lorenz curves discriminate between universal and
non-universal classes of random quantum circuits, and they also detect the
complexity of some non-universal but not classically efficiently simulatable
quantum random circuits. We conclude that majorization can be used as a
indicator of complexity of quantum dynamics, as an alternative to, e.g.,
entanglement spectrum and out-of-time-order correlators (OTOCs).
