  Unitary Designs have become a vital tool for investigating pseudorandomness
since they approximate the statistics of the uniform Haar ensemble. Despite
their central role in quantum information, their relation to quantum chaotic
evolution and in particular to the Eigenstate Thermalization Hypothesis (ETH)
are still largely debated issues. This work provides a bridge between the
latter and $k$-designs through Free Probability theory. First, by introducing
the more general notion of $k$-freeness, we show that it can be used as an
alternative probe of designs. In turn, free probability theory comes with
several tools, useful for instance for the calculation of mixed moments or for
quantum channels. Our second result is the connection to quantum dynamics.
Quantum ergodicity, and correspondingly ETH, apply to a restricted class of
physical observables, as already discussed in the literature. In this spirit,
we show that unitary evolution with generic Hamiltonians always leads to
freeness at sufficiently long times, but only when the operators considered are
restricted within the ETH class. Our results provide a direct link between
unitary designs, quantum chaos and the Eigenstate Thermalization Hypothesis,
and shed new light on the universality of late-time quantum dynamics.
