  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.
