Quantum sensing, quantum networking and communication, and quantum computing have attracted significant attention recently, as these quantum technologies could offer significant advantages over existing technologies. In order to accelerate the commercialization of these quantum technologies, the workforce must be equipped with the necessary skills. Through a qualitative study of the quantum industry, in a series of interviews with 21 U.S. companies carried out in Fall 2019, we describe the types of activities being carried out in the quantum industry, profile the types of jobs that exist, and describe the skills valued across the quantum industry, as well as in each type of job. The current routes into the quantum industry are detailed, providing a picture of the current role of higher education in training the quantum workforce. Finally, we present the training and hiring challenges the quantum industry is facing and how higher education may optimize the important role it is currently playing.