Einstein-Podolsky-Rosen (EPR) steering is an intermediate quantum correlation that lies in between entanglement and Bell non-locality. Its temporal analog, temporal steering, has recently been shown to have applications in quantum information and open quantum systems. Here, we show that there exists a hierarchy among the three temporal quantum correlations: temporal inseparability, temporal steering, and macrorealism. Given that the temporal inseparability can be used to define a measure of quantum causality, similarly the quantification of temporal steering can be viewed as a weaker measure of direct cause and can be used to distinguish between direct cause and common cause in a quantum network.

We propose a measure of quantum steerability, namely a convex steering monotone, based on the trace distance between a given assemblage and its corresponding closest assemblage admitting a local-hidden-state (LHS) model. We provide methods to estimate such a quantity, via lower and upper bounds, based on semidefinite programming. One of these upper bounds has a clear geometrical interpretation as a linear function of rescaled Euclidean distances in the Bloch sphere between the normalized quantum states of: (i) a given assemblage and (ii) an LHS assemblage. For a qubit-qubit quantum state, these ideas also allow us to visualize various steerability properties of the state in the Bloch sphere via the so-called LHS surface. In particular, some steerability properties can be obtained by comparing such an LHS surface with a corresponding quantum steering ellipsoid. Thus, we propose a witness of steerability corresponding to the difference of the volumes enclosed by these two surfaces. This witness (which reveals the steerability of a quantum state) enables one to find an optimal measurement basis, which can then be used to determine the proposed steering monotone (which describes the steerability of an assemblage) optimized over all mutually-unbiased bases.

The Leggett–Garg inequality attempts to classify experimental outcomes as arising from one of two possible classes of physical theories: those described by macrorealism (which obey our intuition about how the macroscopic classical world behaves) and those that are not (e.g., quantum theory). The development of cloud-based quantum computing devices enables us to explore the limits of macrorealism. In particular, here we take advantage of the properties of the programmable nature of the IBM quantum experience to observe the violation of the Leggett–Garg inequality (in the form of a ‘quantum witness’) as a function of the number of constituent systems (qubits), while simultaneously maximizing the ‘disconnectivity’, a potential measure of macroscopicity, between constituents. Our results show that two- and four-qubit ‘cat states’ (which have large disconnectivity) are seen to violate the inequality, and hence can be classified as non-macrorealistic. In contrast, a six-qubit cat state does not violate the ‘quantum witness’ beyond a so-called clumsy invasive-measurement bound, and thus is compatible with ‘clumsy macrorealism’. As a comparison, we also consider un-entangled product states with n = 2, 3, 4 and 6 qubits, in which the disconnectivity is low.