  Catalysts used in quantum resource theories need not be in isolation and
therefore are possibly correlated with external systems, which the agent does
not have access to. Do such correlations help or hinder catalysis, and does the
classicality or quantumness of such correlations matter? To answer this
question, we first focus on the existence of a non-invasively measurable
observable that yields the same outcomes for repeated measurements, since this
signifies macro-realism, a key property distinguishing classical systems from
quantum systems. We show that a system quantumly correlated with an external
system so that the joint state is necessarily perturbed by any repeatable
quantum measurement, also has the same property against general quantum
channels. Our full characterization of such systems called totally quantum
systems, solves the open problem of characterizing tomographically sensitive
systems raised in [Lie and Jeong, Phys. Rev. Lett. 130, 020802 (2023)]. An
immediate consequence is that a totally quantum system cannot catalyze any
quantum process, even when a measure of correlation with its environment is
arbitrarily low. It generalizes to a stronger result, that the mutual
information of totally quantum systems cannot be used as a catalyst either.
These results culminate in the conclusion that, out of the correlations that a
generic quantum catalyst has with its environment, only classical correlations
allow for catalysis, and therefore using a correlated catalyst is equivalent to
using an ensemble of uncorrelated catalysts.
