  This paper concerns the positive part $U^+_q$ of the quantum group
$U_q({\widehat{\mathfrak{sl}}}_2)$. The algebra $U^+_q$ has a presentation
involving two generators that satisfy the cubic $q$-Serre relations. We
recently introduced an algebra $\mathcal U^+_q$ called the alternating central
extension of $U^+_q$. We presented $\mathcal U^+_q$ by generators and
relations. The presentation is attractive, but the multitude of generators and
relations makes the presentation unwieldy. In this paper we obtain a
presentation of $\mathcal U^+_q$ that involves a small subset of the original
set of generators and a very manageable set of relations. We call this
presentation the compact presentation of $\mathcal U^+_q$.
