Recent developments in quantum thermodynamics allowed the definition of thermodynamic quantities at the level of single trajectories based on the quantum jump approach. In particular, stochastic entropy production has been characterized from first principles in the two-point-measurement scheme with the help of environmental monitoring, leading to the derivation of fluctuation theorems extending the second law of thermodynamics to the level of fluctuations .
In this talk I will present a further development of this theory, namely, the derivation of stronger relations for the fluctuations of stochastic entropy production, valid in arbitrary quantum nonequilibrium steady states. These relationships are derived using Martingale stochastic processes, well known in mathematics and finance, but only slightly explored in thermodynamics until now. Martingale theory results in a genuine classical-quantum split of stochastic entropy production in two terms, obeying independent fluctuation theorems, and a set of universal relations for their stopping-time (e.g. fist-passage times, scape times, ...) and Infimum statistics during fixed time-intervals .
 G. Manzano, J. M. Horowitz, and J. M. R. Parrondo, Phys. Rev. X 8, 031037 (2018).
 G. Manzano, R. Fazio, and É. Roldán, Phys. Rev. Lett. 122, 220602 (2019).