Score-based diffusion models have significantly advanced high-dimensional data generation across various domains, by learning a denoising oracle (or score) from datasets. From a Bayesian perspective, they offer a realistic modeling of data priors and facilitate solving inverse problems through posterior sampling. Although many heuristic methods have been developed recently for this purpose, they lack the quantitative guarantees needed in many scientific applications. This work addresses the topic from two perspectives. We first present a hardness result indicating that a generic method leveraging the prior denoising oracle for posterior sampling becomes infeasible as soon as the measurement operator is mildly ill-conditioned.