Recent studies of the quark-matter equation of state constrained by neutron-star mass-radius measurements strongly supports a smooth, continuous crossover from nuclear matter to the possible quark matter at several times nuclear matter densities. Aside from favoring quark-hadron continuity, the required stiffness of the equation of state also indicates strong diquark pairing and vector repulsion that are significantly stronger than the traditionally considered Fierz-transformed four-quark couplings in phenomenological quark models.
In this talk I will discuss how we can understand cold, dense quark matter in several aspects deeply related to those findings. Firstly, I will talk about the realization of quark-hadron continuity from the perspective of the low-lying Nambu-Goldstone modes due to spontaneous chiral symmetry breaking by coexisting chiral and diquark condensates in quark matter. As density increases, they gradually evolve from $\bar qq$ modes into $\bar q\bar q qq$ modes with their properties (mass, decay constants and coupling to quarks) changing continuously. Next, I will show that quark matter in a color-flavor-locked color superfluid can be described entirely in a manifestly gauge-invariant manner. Thus a theory of quarks interacting with collective mesonic modes and gluons can be completely transformed into a hadron-like theory, where gauge invariant baryon-like fermions interact through mesonic currents. Lastly, I will discuss how we may begin to understand the strong vector repulsions in quark matter from non-perturbative gluon exchange, thus bridging the theories of sub-GeV QCD to quark matter inside neutron stars.