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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. I will explain how the complexity rigorously defines the optimal compression of a many-body state in the Fock space, thus determining the minimum computational and physical resources required to represent it. We uncover a remarkable connection between the Fock complexity and the single-particle entanglement entropy, underlying the complexity scaling at fixed filling. In particular, 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.