  Using a random matrix approach, combined with the ergodicity hypothesis, we
derive an analytical expression for the time-averaged quantum Fisher
information (QFI) that applies to all quantum chaotic systems governed by
Dyson's ensembles. Our approach integrates concepts of randomness, multipartite
entanglement and quantum chaos. Furthermore, the QFI proves to be highly
dependent on the dimension of the Krylov space confining the chaotic dynamics:
it ranges from $N^2/3$ for $N$ qubits in the permutation-symmetric subspace
(e.g. for chaotic kicked top models with long-range interactions), to $N$ when
the dynamics extend over the full Hilbert space with or without bit reversal
symmetry or parity symmetry (e.g. in chaotic models with short-range Ising-like
interactions). In the former case, the QFI reveals multipartite entanglement
among $N/3$ qubits. Interestingly this result can be related to isotropic
substructures in the Wigner distribution of chaotic states and demonstrates the
efficacy of quantum chaos for Heisenberg-scaling quantum metrology. Finally,
our general expression for the QFI agrees with that obtained for random states
and, differently from out-of-time-order-correlators, it can also distinguish
chaotic from integrable unstable spin dynamics.
