We investigate the coefficients of the highest and lowest terms (also called
the head and the tail) of the colored Jones polynomial and show that they
stabilize for closures of alternating braids. We also see that for closures of
positive braids, the lowest terms are trivial. We do this by using the quantum
determinant expression for the colored Jones polynomial introduced by Vu Huynh
and Thang L\^{e} and deriving a combinatorial description of this quantum
determinant in terms of walks along the braid.