  The most obvious obstacle behind a direct test of Quantum Gravity (QG) is its
energy scale ($10^{19}$ GeV), which remains well outside of any human made
machine. The next best possible approach is to provide indirect tests on
effective theories of QG which can be performed in a lower energy scale. This
paper is aimed in this direction, and shows a promising path to test the
existence of the fundamental minimal length scale of Nature by measuring the
dispersion of free, large molecular wave-packets. The existence of the minimal
length is believed to be the reason for a modified commutation relationship
between the position and momentum operators and, in this paper, we show that
such a modification of the commutator has a profound effect on the dispersion
rate of free wave-packets, and precise measurement on the broadening times of
large molecular wave-packets (such as $C_{60}$, $C_{176}$ and large organic
molecules) provide a promising path for an indirect test of quantum gravity, in
a laboratory setting.
