How do entangled states facilitate the implementation of quantum algorithms, such as Grover's and Shor's algorithms?

Entangled states provide the parallelism and interference effects necessary for quantum algorithms to outperform classical counterparts. In Grover’s algorithm, entanglement enables the superposition of multiple search states and their simultaneous evolution, leading to quadratic speedup in search tasks. In Shor’s algorithm, entanglement is essential for the quantum Fourier transform and the factorization of large numbers, achieving exponential speedup in solving problems like integer factorization.