  The dynamics of quantum systems strongly depends on the local structure of
the Hamiltonian. For short-range interacting systems, the well-known
Lieb-Robinson bound defines the effective light cone with an exponentially
small error with respect to the spatial distance, whereas we can obtain only
polynomially small error for distance in long-range interacting systems. In
this paper, we derive a qualitatively new bound for quantum dynamics by
considering how many spins can correlate with each other after time evolution.
Our bound characterizes the number of spins which support the many-body
entanglement with exponentially small error and is valid for large class of
Hamiltonians including long-range interacting systems. To demonstrate the
advantage of our approach in quantum many-body systems, we apply our bound to
prove several fundamental properties which have not be derived from the
Lieb-Robinson bound.
