  The Bell inequality, and its substantial experimental violation, offers a
seminal paradigm for showing that the world is not in fact locally realistic.
Here, going beyond the scope of Bell's inequality on physical states, we show
that quantum teleportation can be used to quantitatively characterize quantum
correlations of physical processes. The validity of the proposed formalism is
demonstrated by considering the problem of teleportation through a linear
three-node quantum network. A hierarchy is derived between the Bell
nonlocality, nonbilocality, steering and nonlocality-steering hybrid
correlations based on a process fidelity constraint. The proposed framework can
be directly extended to reveal the nonlocality structure of teleportation
through any linear many-node quantum network. The formalism provides a faithful
identification of quantum teleportation and demonstrates the use of
quantum-information processing as a means of quantitatively discriminating
quantum correlations.
