  How much information a fermionic state contains? To address this fundamental
question, we define the complexity of a particle-conserving many-fermion state
as the entropy of its Fock space probability distribution, minimized over all
Fock representations. The complexity characterizes the minimum computational
and physical resources required to represent the state and store the
information obtained from it by measurements. Alternatively, the complexity can
be regarded a Fock space entanglement measure describing the intrinsic
many-particle entanglement in the state. We establish universal lower bound for
the complexity in terms of the single-particle correlation matrix eigenvalues
and formulate a finite-size complexity scaling hypothesis. Remarkably,
numerical studies on interacting lattice models suggest a general
model-independent complexity hierarchy: ground states are exponentially less
complex than average excited states which, in turn, are exponentially less
complex than generic states in the Fock space. Our work has fundamental
implications on how much information is encoded in fermionic states.
