  The investigation of many-body interactions holds significant importance in
both quantum foundations and information. Hamiltonians coupling multiple
particles at once, beyond others, can lead to a faster entanglement generation,
multiqubit gate implementation and improved error correction. As an increasing
number of quantum platforms enable the realization of such physical settings,
it becomes interesting to study the verification of many-body interaction
resources. In this work, we explore the possibility of higher-order couplings
detection through the quantum Fisher information. For a family of symmetric and
translationally invariant $k$-body Ising-like Hamiltonians, we derive the
bounds on the quantum Fisher information in product states. Due to its ordering
with respect to the order of interaction, we demonstrate the possibility of
detecting many-body couplings for a given Hamiltonian from the discussed family
by observing violations of an appropriate bound. As a possible extension to
these observations, we further analyse an example concerning the three-body
interaction detection in the XY model.
