Characterizing the complete hierarchy of correlations in an $n$-party system

A characterization of the complete correlation structure in an $n$-party system is proposed in terms of a series of $(k,n)$ threshold classical secret sharing protocols ($2le kle n$). The total correlation is shown to be the sum of independent correlations of 2-, 3-,$...$, $n$-parties. Our result unifies several earlier scattered works, and shines new light at the important topic of multi-party quantum entanglement. As an application, we explicitly construct the hierarchy of correlations in an $n$-qubit graph state.