Universal two-time correlations, out-of-time-ordered correlators and Leggett-Garg inequality violation by edge Majorana fermion qubits.

Majorana fermion chains (MFC) can be realized in a variety of condensed matter systems ranging from quantum spin chains to hybrid semiconductor-superconductor nanostructures. They support exotic quantum states with strong non-local spatial properties that have been extensively studied. However, the temporal correlations have been mostly ignored so far. We present analytical as well as numerical results of two-time correlations (TTCs) of Majorana edge localized fermions for open chains based on Ising or Kitaev models. We uncover universal relationships between the short time TTC decay and a particular family of out-of-time-ordered correlators which provide efficient and direct experimental alternatives to the estimation of basic Hamiltonian parameters. Furthermore, we identify some edge TTCs displaying long-time saturation values which behave as order-parameter indicators. Finally, we find that violations of Leggett-Garg inequalities can assess the topological-trivial phase transition by looking at different qubits formed by pairing local and non-local edge Majorana fermions. This work lays the foundation for further exploration of temporal correlations in more complex scenarios such as 2D and 3D topological systems.