Complex networks have become increasingly popular for modeling various real-world phenomena. Realistic generative network models are important in this context as they avoid privacy concerns of real data and simplify complex network research regarding data sharing, reproducibility, and scalability studies. \emph{Random hyperbolic graphs} are a well-analyzed family of geometric graphs. Previous work provided empirical and theoretical evidence that this generative graph model creates networks with non-vanishing clustering and other realistic features. However, the investigated networks in previous applied work were small, possibly due to the quadratic running time of a previous generator. In this work we provide the first generation algorithm for these networks with subquadratic running time. We prove a time complexity of $O((n^{3/2}+m) \log n)$ with high probability for the generation process. This running time is confirmed by experimental data with our implementation. The acceleration stems primarily from the reduction of pairwise distance computations through a polar quadtree, which we adapt to hyperbolic space for this purpose. In practice we improve the running time of a previous implementation by at least two orders of magnitude this way. Networks with billions of edges can now be generated in a few minutes. Finally, we evaluate the largest networks of this model published so far. Our empirical analysis shows that important features are retained over different graph densities and degree distributions.