Big data refers to group of information that are so huge and bulky that ordinary database management tools have trouble capturing, storing, sharing and managing the information. With the arrival of big data arises the threat of superior security violations as data volume enlarges. Since many years cryptography has evolved to such an extent that breaking it through simple techniques is impossible. Neural and Quantum cryptography are the two emerging branches of cryptography. Neural cryptography incorporates the idea of artificial neural network with classical cryptographic algorithms; whereas quantum cryptography incorporates phenomenon of quantum physics for secure communications. This paper covers a brief introduction about big data, classical, quantum and neural cryptography and finally the comparative analysis between quantum and neural cryptography and conclusion based on the analysis. The comparison is based on the methodologies used in both the cryptographic techniques, use of these cryptographies in security of big data and their advantages and disadvantages.