  We present a progress on local computation algorithms for two coloring of
$k$-uniform hypergraphs. We focus on instances that satisfy strengthened
assumption of Local Lemma of the form $2^{1-\alpha k} (\Delta+1) e < 1$, where
$\Delta$ is the bound on the maximum edge degree of the hypergraph. We discuss
how previous works on the subject can be used to obtain an algorithm that works
in polylogarithmic time per query for $\alpha$ up to about $0.139$. Then, we
present a procedure that, within similar bounds on running time, solves wider
range of instances by allowing $\alpha$ at most about $0.227$.
