The paper discusses the computational power of quantum Turing machines (QTMs) and addresses the question of whether quantum computers can solve NP-complete problems efficiently. It proves that, relative to a uniformly random oracle, the class NP cannot be solved on a quantum Turing machine in time \(o(2^{n/2})\), and the class NP ∩ co-NP cannot be solved in time \(o(2^{n/3})\). These results are tight, as Grover's algorithm shows that NP can be solved in time \(O(2^{n/2})\) relative to any oracle. The paper also explores the limitations of simulating nondeterminism on QTMs and provides a method to modify any BQP machine into a "tidy" BQP machine, ensuring that the final superposition consists almost entirely of a tape configuration containing just the input and the single-bit answer. This allows BQP machines to be used as subroutines, leading to the conclusion that BQP^BQP = BQP. The paper concludes by discussing the relevance of these results and the challenges in simulating nondeterminism on QTMs.The paper discusses the computational power of quantum Turing machines (QTMs) and addresses the question of whether quantum computers can solve NP-complete problems efficiently. It proves that, relative to a uniformly random oracle, the class NP cannot be solved on a quantum Turing machine in time \(o(2^{n/2})\), and the class NP ∩ co-NP cannot be solved in time \(o(2^{n/3})\). These results are tight, as Grover's algorithm shows that NP can be solved in time \(O(2^{n/2})\) relative to any oracle. The paper also explores the limitations of simulating nondeterminism on QTMs and provides a method to modify any BQP machine into a "tidy" BQP machine, ensuring that the final superposition consists almost entirely of a tape configuration containing just the input and the single-bit answer. This allows BQP machines to be used as subroutines, leading to the conclusion that BQP^BQP = BQP. The paper concludes by discussing the relevance of these results and the challenges in simulating nondeterminism on QTMs.