  We extend a result of Thistlethwaite [17, Theorem 1(iv)] on the structure of
the Jones polynomial of alternating links to the wider class of
quasi-alternating links. In particular, we prove that the Jones polynomial of
any prime quasi-alternating link that is not a $(2,n)$-torus link has no gap.
As an application, we show that the differential grading of the Khovanov
homology of any prime quasi-alternating link that is not a $(2,n)$-torus link
has no gap. Also, we show that the determinant is an upper bound for the
breadth of the Jones polynomial for any quasi-alternating link. Finally, we
prove that the Jones polynomial of any non-prime quasi-alternating link $L$ has
more than one gap if and only if $L$ is a connected sum of Hopf links.
