Is there a hash or string that represents a stockfish evaluation of a position to a certain depth? For example for a given FEN, can you analyze to a depth of say 10 and get a hash to represent that position AND depth of analysis? That would allow you to come back later and, if wanted, continue analysis to a greater depth at a later point.
I doubt that can be possible.
An analysis to a certain depth contains the generated position tree and the calculated values, which is quickly becoming a very large amount of data.
For depth 10, you probably need several megabytes of memory to represent the analysis result in a way that it could be used to continue searching deeper. There is simply no way to compress this into a small hash or anything else.
It is definitely impossible. Stockfish uses a game-tree search with pruning and evaluation heuristics. If it has searched to a certain 'depth', it means that it has built (in memory) a game tree to that depth excluding many pruned branches. To deepen a search, Stockfish would not only have to continue the search to subsequent moves past the end of each originally searched path, but also have to check many heuristically pruned branches (which may include those that were pruned as per alpha-beta pruning, since the evaluations may change). Doing so would require knowing practically the entire game-tree. There is thus absolutely no way to compress this into a short string, short of knowing a simple and optimal strategy for playing chess!