  We develop generalized bounds for quantum single-parameter estimation
problems for which the coupling to the parameter is described by intrinsic
multi-system interactions. For a Hamiltonian with $k$-system
parameter-sensitive terms, the quantum limit scales as $1/N^k$ where $N$ is the
number of systems. These quantum limits remain valid when the Hamiltonian is
augmented by any parameter independent interaction among the systems and when
adaptive measurements via parameter-independent coupling to ancillas are
allowed.
