Explain the comparative power of matrix- vs. scalar-valued test supermartingales

Provide a theoretical explanation for the empirical differences in power observed between matrix-valued and scalar-valued test supermartingales when testing the same sequential null hypotheses, clarifying when and why matrix-based methods are more powerful.

Background

The empirical section compares matrix-valued and scalar-valued test supermartingales, finding scenarios where matrix-based procedures are more powerful (particularly under non-commuting observations). A theoretical account of these phenomena is lacking, and understanding it could guide the choice of methods in practice.