Sun and Tauraso conjectured that for any positive integer $a$ we have
$$\sum_{k=0}^{3^a-1}\binom{2k}{k}=0 (mod 3^{2a})$$ and furthermore
$$3^{-2a}}\sum_{k=0}^{3^a-1}\binom{2k}k=1 (mod 3).$$ Recently a $q$-analogue of
the first congruence was conjectured by Guo and Zeng. In this paper we prove
both conjectures.