Methods and apparatus for using characterized devices such as memories. In one embodiment, characterized memories are associated with a range of performances over a range of operational parameters. The characterized memories can be used in conjunction with a solution density function to optimize memory searching. In one exemplary embodiment, a cryptocurrency miner can utilize characterized memories to generate memory hard proof-of-work (POW). The results may be further validated against general compute memories; such that only valid solutions are broadcasted to the mining community. In one embodiment, the validation mechanism is implemented for a plurality of searching apparatus in parallel to provide a more distributed and efficient approach. Various other applications for characterized memories are also described in greater detail herein (e.g., blockchain, social media, machine learning, probabilistic applications and other error-tolerant applications).