Quantum classification and hypothesis testing are two tightly related subjects, the main difference being that the former is data driven: how to assign to quantum states $ρ(x)$ the corresponding class $c$ (or hypothesis) is learnt from examples during training, where $x$ can be either tunable experimental parameters or classical data "embedded" into quantum states. Does the model generalize? This is the main question in any data-driven strategy, namely the ability to predict the correct class even of previously unseen states. Here we establish a link between quantum machine learning classification and quantum hypothesis testing (state and channel discrimination) and then show that the accuracy and generalization capability of quantum classifiers depend on the (Rényi) mutual informations $I(C{:}Q)$ and $I_2(X{:}Q)$ between the quantum state space $Q$ and the classical parameter space $X$ or class space $C$. Based on the above characterization, we then show how different properties of $Q$ affect classification accuracy and generalization, such as the dimension of the Hilbert space, the amount of noise, and the amount of neglected information from $X$ via, e.g., pooling layers. Moreover, we introduce a quantum version of the Information Bottleneck principle that allows us to explore the various tradeoffs between accuracy and generalization. Finally, in order to check our theoretical predictions, we study the classification of the quantum phases of an Ising spin chain, and we propose the Variational Quantum Information Bottleneck (VQIB) method to optimize quantum embeddings of classical data to favor generalization.