Persistent homology naturally addresses the multi-scale topological characteristics of the
large-scale structure as a distribution of clusters, loops, and voids. We apply this tool to the
dark matter halo catalogs from the Quijote simulations, and build a summary statistic for
comparison with the joint power spectrum and bispectrum statistic regarding their information
content on cosmological parameters and primordial non-Gaussianity. Through a Fisher analysis, we
find that constraints from persistent homology are tighter for 8 out of the 10 parameters by
margins of 13–50%. The complementarity of the two statistics breaks parameter degeneracies,
allowing for a further gain in constraining power when combined. We run a series of consistency
checks to consolidate our results, and conclude that our findings motivate incorporating
persistent homology into inference pipelines for cosmological survey data.