Social interactions are composite, involve different communication layers and evolve in time. However, a rigorous analysis of the whole complexity of social networks has been hindered so far by lack of suitable data. Here we consider both the multi-layer and dynamic nature of social relations by analysing a diverse set of empirical temporal multiplex networks. We focus on the measurement and characterization of inter-layer correlations to investigate how activity in one layer affects social acts in another layer. We define observables able to detect when genuine correlations are present in empirical data, and single out spurious correlation induced by the bursty nature of human dynamics. We show that such temporal correlations do exist in social interactions where they act to depress the tendency to concentrate long stretches of activity on the same layer and imply some amount of potential predictability in the connection patterns between layers. Our work sets up a general framework to measure temporal correlations in multiplex networks, and we anticipate that it will be of interest to researchers in a broad array of fields.