Title: Quantum Alphatron. Abstract: Finding provably efficient algorithms for learning neural networks is a fundamental challenge in the theory of machine learning. The Alphatron of Goel and Klivans is the first provably efficient algorithm for learning neural networks with more than one nonlinear layer. The algorithm succeeds with any distribution on the $n$-dimensional unit ball and without any assumption on the structure of the network. In this work, we refine the original Alphatron by a pre-computing phase for its most time-consuming part, the evaluation of the kernel function. This refined algorithm improves the run time of the original Alphatron, while retaining the same learning guarantee. Based on the refined algorithm, we quantize the pre-computing phase with provable learning guarantee in the fault-tolerant quantum computing model. In a well-defined learning model, this quantum algorithm is able to provide a quadratic speedup in the data dimension $n$. In addition, we discuss the second type of speedup, quantizing the evaluation of the gradient in the stochastic gradient descent procedure. Our work contributes to the study of quantum learning with kernels and from samples.