Comparing the quantum witness, the entropic Leggett-Garg inequality and the NCGD.

In this paper, we investigate the violation of the quantum witness, the entropic Leggett-Garg inequality (LGI) and the no-coherence-generating-and-detecting (NCGD) dynamics, under projective and coarsening measurements. We consider a qubit in the three scenarios: coherent dynamics, in the presence of dissipation, and in the presence of dephasing. For the pure qubit, we find that in the case of the projective measurement, the non-violation conditions of the quantum witness and the NCGD are the same; while the non-violation conditions of the entropic LGI and the quantum witness do not contain each other, i.e., a suitable conjunction of the quantum witness and the entropic LGI may be better for testing macrorealism. Also, for the pure qubit with coarsening measurement similar results can be obtained. For the dissipative qubit with projective measurement, the quantum witness and the NCGD can be both violated for a wider parameter regime than the entropic LGI. For the dissipative qubit with coarsening measurement, the violation of the NCGD is the most robust compared to the quantum witness and the entropic LGI. For the dephasing qubit with projective and coarsening measurements, the relationship among the quantum witness, the entropic LGI and the NCGD is similar to that of the pure qubit. In addition, we find that for pure, dissipative and dephasing qubits, the robustness of the coarsening measurement in final resolution is more vulnerable than that of the coarsening measurement in reference for the entropic LGI.