  Complexity of dynamics is at the core of quantum many-body chaos and exhibits
a hierarchical feature: higher-order complexity implies more chaotic dynamics.
Conventional ergodicity in thermalization processes is a manifestation of the
lowest order complexity, which is represented by the eigenstate thermalization
hypothesis (ETH) stating that individual energy eigenstates are thermal. Here,
we propose a higher-order generalization of the ETH, named the $ k $-ETH ($
k=1,2,\dots $), to quantify higher-order complexity of quantum many-body
dynamics at the level of individual energy eigenstates, where the lowest order
ETH (1-ETH) is the conventional ETH. As a non-trivial contribution of the
higher-order ETH, we show that the $ k $-ETH with $ k\geq 2 $ implies a
universal behavior of the $ k $th Renyi entanglement entropy of individual
energy eigenstates. In particular, the Page correction of the entanglement
entropy originates from the higher-order ETH, while as is well known, the
volume law can be accounted for by the 1-ETH. We numerically verify that the
2-ETH approximately holds for a nonintegrable system, but does not hold in the
integrable case. To further investigate the information-theoretic feature
behind the $ k $-ETH, we introduce a concept named a partial unitary $ k
$-design (PU $ k $-design), which is an approximation of the Haar random
unitary up to the $ k $th moment, where partial means that only a limited
number of observables are accessible. The $ k $-ETH is a special case of a PU $
k $-design for the ensemble of Hamiltonian dynamics with random-time sampling.
In addition, we discuss the relationship between the higher-order ETH and
information scrambling quantified by out-of-time-ordered correlators. Our
framework provides a unified view on thermalization, entanglement entropy, and
unitary $ k $-designs, leading to deeper characterization of higher-order
quantum complexity.
