Suppose A,B are nxn matrices, and x is an nx1 vector. Need to compute M1M2...Mm x where each Mi is either A or B. The straightforward approach is to start multiplying from the right, which takes O(n^2 m) operations. Is it possible to have an O(n^2 m) algorithm that solves the problem above, but has a lower time complexity than the baseline?

The lower bound is O(n^2+m) because that's how long it takes to input the problem. If you could improve on the standard approach, you could solve the problem of inference in Markov-1 hidden markov model with binary observations faster than the Forward algorithm

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