  In terms of the $q$-Saalsch\"{u}tz identity and the Chinese remainder theorem
for coprime polynomials, we establish some $q$-supercongruences modulo the
third power of a cyclotomic polynomial. In particular, we give a $q$-analogue
of a formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773--808].
