Quantum computing has been shown to be more powerful than classical probabilistic computing, as demonstrated by results such as Shor's algorithm for factoring and discrete logarithms. However, this paper proves that relative to a random oracle, quantum Turing machines cannot solve NP in time o(2^{n/2}) and NP ∩ co-NP in time o(2^{n/3}). These results are tight, as Grover's algorithm shows that NP can be solved in O(2^{n/2}) time on a quantum computer. The paper also discusses the limitations of quantum computing in solving NP-complete problems, emphasizing that no black-box approach can exploit quantum-mechanical features to solve these problems efficiently. It further explores the use of quantum Turing machines as subroutines, showing that BQP^BQP = BQP. The paper concludes that quantum computers are inherently more powerful than classical computers in a model-independent way, but that there are fundamental limits to their computational power when solving NP-type problems relative to random oracles.Quantum computing has been shown to be more powerful than classical probabilistic computing, as demonstrated by results such as Shor's algorithm for factoring and discrete logarithms. However, this paper proves that relative to a random oracle, quantum Turing machines cannot solve NP in time o(2^{n/2}) and NP ∩ co-NP in time o(2^{n/3}). These results are tight, as Grover's algorithm shows that NP can be solved in O(2^{n/2}) time on a quantum computer. The paper also discusses the limitations of quantum computing in solving NP-complete problems, emphasizing that no black-box approach can exploit quantum-mechanical features to solve these problems efficiently. It further explores the use of quantum Turing machines as subroutines, showing that BQP^BQP = BQP. The paper concludes that quantum computers are inherently more powerful than classical computers in a model-independent way, but that there are fundamental limits to their computational power when solving NP-type problems relative to random oracles.