Fast sparse boolean matrix chain product
This was too long to be a comment -- I wonder if those matrices have structure that make them behave differently from random matrices. Products of random sparse matrices go to zero or become non-sparse quickly.
Here's a simple experiment -- take 200 random binary 50x50 matrices, and plot number of non-zeros as a function of number of matrices multiplied. Plots below show standard deviation over 2000 runs. First plot is for 2% sparsity, second plot is for 3%

(source: yaroslavvb.com)

(source: yaroslavvb.com)
this took 3 minutes on my laptop using standard matrix multiplication