Hierarchy of efficiently computable and faithful lower bounds to quantum discord

Quantum discord expresses a fundamental non-classicality of correlations that is more general than entanglement, but that, in its standard definition, is not easily evaluated. We derive a hierarchy of computationally efficient lower bounds to the standard quantum discord. Every non-trivial element of the hierarchy constitutes by itself a valid discord-like measure, based on a fundamental feature of quantum correlations: their lack of shareability. Our approach emphasizes how the difference between entanglement and discord depends on whether shareability is intended as a static property or as a dynamical process.