Generative adversarial networks (GANs) have been extremely successful in generating samples, from seemingly high dimensional probability measures. However, these methods struggle to capture the temporal dependence of joint probability distributions induced by time-series data. Furthermore, long time-series data streams hugely increase the dimension of the target space, which may render generative modeling infeasible. To overcome these challenges, we integrate GANs with mathematically principled and efficient path feature extraction called the signature of a path. The signature of a path is a graded sequence of statistics that provides a universal description for a stream of data, and its expected value characterizes the law of the time-series model. In particular, we a develop new metric, (conditional) Sig-W1, that captures the (conditional) joint law of time series models, and use it as a discriminator. The signature feature space enables the explicit representation of the proposed discriminators which alleviates the need for expensive training. Furthermore, we develop a novel generator, called the conditional AR-FNN, which is designed to capture the temporal dependence of time series and can be efficiently trained. We validate our method on both synthetic and empirical datasets and observe that our method consistently and significantly outperforms state-of-the-art benchmarks with respect to measures of similarity and predictive ability.