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Quantum generators are crucial objects for understanding quantum systems. They encode all the information required to predict the evolution of closed (Hamiltonians) and open memoryless dynamics (Lindbladians). Additionally, they offer structural insight into the noise affecting experimental implementations of quantum processes. In this talk, we will discuss the definition of continuous-time random unitary evolutions characterized by stochastic Hamiltonians, showing mixing properties and efficient convergence to the uniform measure over the unitary group. We will then consider the problem of embedding a quantum map into a Markovian evolution, presenting a scheme to extrapolate the full description of the Lindbladian that is the best fit for the tomographic measurements of any (noisy) quantum channel. Finally, we will introduce a new bound on quantum operators generated by arbitrary time-dependent Hamiltonians by leveraging a correspondence with binary trees structures. References: [1] E. Onorati et al., ‘Mixing properties of stochastic quantum Hamiltonians’, https://arxiv.org/abs/1606.01914 [2] E. Onorati et al., ‘Fitting quantum noise models to tomographic data’, https://arxiv.org/abs/2103.17243