We show that the new quantum extension of Renyi's α-relative entropies, introduced recently by Muller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203, (2013), and Wilde, Winter, Yang, Commun. Math. Phys. 331, (2014), have an operational interpretation in the strong converse problem of quantum hypothesis testing. Together with related results for the direct part of quantum hypothesis testing, known as the quantum Hoeffding bound, our result suggests that the operationally relevant definition of the quantum Renyi relative entropies depends on the parameter α: for α<1, the right choice seems to be the traditional definition, whereas for α>1 the right choice is the newly introduced version. As a sideresult, we show that the new Renyi α-relative entropies are asymptotically attainable by measurements for α>1, and give a new simple proof for their monotonicity under completely positive trace-preserving maps.