  We find a novel confinement mechanism in the two-flavor dense quark matter
proposed recently, that consists of the 2SC condensates and the $P$-wave
diquark condensates of $d$-quarks. This quark matter exhibiting color
superconductivity as well as superfluidity is classified into two phases;
confined and deconfined phases of vortices. We establish that the criterion of
the confinement is color neutrality of Aharonov-Bohm (AB) phases: vortices
exhibiting color non-singlet AB phases are confined by the so-called AB defects
to form color-singlet bound states. In the deconfined phase, the most stable
vortices are non-Abelian Alice strings, which are superfluid vortices with
fractional circulation and non-Abelian color magnetic fluxes therein,
exhibiting color non-singlet AB phases. On the other hand, in the confined
phase, these non-Abelian vortices are confined to either a baryonic or mesonic
bound state in which constituent vortices are connected by AB defects. The
baryonic bound state consists of three non-Abelian Alice strings with different
color magnetic fluxes with the total flux canceled out connected by a domain
wall junction, while the mesonic bound state consists of two non-Abelian Alice
strings with the same color magnetic fluxes connected by a single domain wall.
Interestingly, the latter contains a color magnetic flux in its core, but this
can exist because of color neutrality of its AB phase.
