Temporal correlation, as a resource for quantum information tasks such as quantum computation, randomness generation and secure key distribution, has attracted wide interests. In this paper, we study the temporal correlation by comparing the Leggett–Garg inequality (LGI), the Wigner form of LGI (WLGI), the entropic LGI, the no-signaling-in-time (NSIT) condition and the no-coherence-generating-and-detecting (NCGD) evolution condition in a large spin system for the projective measurement and the temporal coarsening measurement. Furthermore, we consider two initial states: \(m_\textrm{zint} = -j\) and \(m_\textrm{zint} = -\frac{1}{2}\) in this paper. For the projective measurement and the initial state \(m_\textrm{zint} = -j\) , as the system becomes larger, the violation conditions of the LGI, the WLGI, the NSIT condition and the NCGD do not change, and the range of the violation condition of the entropic LGI becomes wider. In addition, the WLGI, the NSIT condition and the NCGD can be violated for a wider parameter regime than the LGI and entropic LGI with the projective measurement for the different total spin angular momentumj, in the initial state \(m_\textrm{zint} = -j\) . For the projective measurement and the initial state \(m_\textrm{zint} =-\frac{1}{2}\) , as the system becomes larger, the violation conditions of the NSIT condition and the NCGD do not change, and the ranges of the violation conditions of the LGI, the WLGI, the entropic LGI become narrower. And for some different total spin angular momentum j, in the case of the projective measurement and the initial state \(m_\textrm{zint} =-\frac{1}{2}\) , we find that the NSIT condition and the NCGD can be violated for a wider parameter regime than the WLGI, the LGI and entropic LGI. For the coarsening time measurement and the initial state of \(m_\textrm{zint} =-j\) , the ranges of parameters showing quantum-classical discrepancies of the LGI, the WLGI, the entropic LGI and the NCGD become wider, as the value of the total spin angular momentum j increases. However, for the initial state of \(m_\textrm{zint} =-\frac{1}{2}\) , the ranges of parameters showing quantum-classical discrepancies of the LGI, the WLGI, the entropic LGI and the NCGD become narrower, as j increases. In addition, for the initial state both in \(m_\textrm{zint} = -j\) and \(m_\textrm{zint} =-\frac{1}{2}\) , we find that under the temporal coarsening measurement, the NCGD is the most robust, and the robustness of the entropic LGI is the most vulnerable.
