The Grover algorithm for unstructured search and the Shor algorithm for factoring numbers are two important innovations that have demonstrated the quantum computational advantage, and they are covered in detail in this chapter. We study the mathematics of Grover’s amplitude amplification, which accelerates a quadratic search on a database, and Shor’s exponential speedup of cracking RSA cryptography, which involves quantum Fourier transforms on the one hand and period detection on the other. The book chapter highlights several noteworthy implementation considerations, including qubit scalability constraints, quantum error correction overhead, and limitations on the design of oracles. We also hypothesize about the revolutionary impact of hybrid quantum–classical algorithms and the standardization of post-quantum cryptography for machines of the NISQ era. These algorithms demonstrate how, despite present hardware constraints, quantum computing holds promise for expanding the computational boundaries in artificial intelligence, optimization, and even cryptography. The societal ramifications and upcoming research on fault-tolerant systems are the main topics of the critique.