  Our polycrystalline sample study on the Yb-based chalcogenide spinels
$A$Yb$_2$$X_4$ ($A =$ Cd, Mg, $X =$ S, Se) has revealed frustrated magnetism
due to the antiferromagnetically coupled Heisenberg spin on the pyrochlore
lattice. Our crystal electric field analysis indicates the Yb ground state has
nearly Heisenberg spins with a strong quantum character of the ground doublet.
All the materials exhibit an antiferromagnetic order at 1.4-1.8 K, much lower
temperature than the antiferromagnetic exchange coupling scale of $\sim 10$ K.
The magnetic specific heat $C_{\rm M}$ shows a $T^{3}$ dependence, indicating
the gapless feature in the Yb-based chalcogenide spinels. The magnetic entropy
change much smaller than $R$ ln 2 below the N\'{e}el temperature and the small
local magnetic field estimated from the $\mu$SR measurements strongly suggest
the significantly reduced size of the ordered moment in comparison with the
bare moment size 1.33 $\mu_{\rm B}$/Yb supporting strong fluctuations in the
commensurate and incommensurate ordered states in CdYb$_2$S$_4$ and
MgYb$_2$S$_4$, respectively.
