Topological quantum neural networks (TQNN) are novel computational tools that are written within the very same language of topological quantum physics. They can be mapped onto spin-networks, their operation being described in terms of topological quantum field ?theory (TQFT). The new framework enables us to understand deep neural networks as a subcase of TQNN, in the sense that they emerge ?as the semiclassical limit of TQNN, and to rephrase several ?key-concepts of machine learning exploiting the terminology of TQFT. ?Our framework provides as well a working hypothesis for ?understanding the generalization behavior of DNN, relating it to the ?topological features of the graphs structures involved. From the computational? side, an algorithm for the physical scalar product ?between spin-network quantum states with hexagonal shape has been recovered,? together with M. Lulli and E. Zappala, in order to ?implement classifiers, using classical and quantum recoupling. ?Results hitherto obtained seem very promising for using TQNNs as an ?image classifier.? The theoretical framework can be then leveraged to include out-of-equilibrium dynamics. At this purpose, exploiting ?theory modifications of TQFTs that provide specific realizations of the Einstein-Hilbert action in D dimensions, a novel approach can be used to investigate the renormalization group equations from a ?geometrical perspective. The Stochastic Ricci-Flow (SRF) entails the ?breakdown of the diffeomorphism invariance away from equilibrium, ?where the classical theory is recovered.? Phenomenological ?applications to cosmology and astro-particle physics, and recent ?results for quantum decoherence and the measurement problem will be discussed. Both the algorithms rooted in TQNN, and their ?out-of-equilibrium extension, are tailored to be applied to several context in particle physics and condensed matter. This peculiarity ?of our framework enables the study of material implementations of quantum logic circuits associated with TQNNs.