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Quantum computing is an emerging field that uses the concepts of quantum mechanics to outperform classical computers. Quantum computing finds plethora of applications in the 5G and Beyond networks. It can process data at an exponential rate, which can address numerous business and scientific challenges. In this paper, we provide a detailed review of the field, starting from the most important applications of quantum computers and then diving into the future of cryptography. The major applications of quantum computing include unstructured search, quantum simulation, and optimization. It can also provide improvements in terms of speed and accuracy for some existing technologies, such as machine learning. These technologies have numerous applications in 5G and beyond networks. However, due to such abilities of quantum computing, it can also pose a serious risk to many existing security systems, especially the asymmetric key cryptography schemes. The risk of quantum computing has also influenced the mobile broadband standards to move from a symmetric key cryptography techniques to PKI-based trust model. We also discuss in detail various alternate cryptosystems based on mathematical problems that are believed to be hard even for a quantum computer to solve. In parallel, we discuss the developments in the field of quantum key distribution that makes use of quantum phenomenon to develop a quantum-resistant crypto systems. Such quantum-resistant systems have a great potential in provisioning secure 5G and beyond networks.