  Fix a poset $Q$ on $\{x_1,\ldots,x_n\}$. A $Q$-Borel monomial ideal $I
\subseteq \mathbb{K}[x_1,\ldots,x_n]$ is a monomial ideal whose monomials are
closed under the Borel-like moves induced by $Q$. A monomial ideal $I$ is a
principal $Q$-Borel ideal, denoted $I=Q(m)$, if there is a monomial $m$ such
that all the minimal generators of $I$ can be obtained via $Q$-Borel moves from
$m$. In this paper we study powers of principal $Q$-Borel ideals. Among our
results, we show that all powers of $Q(m)$ agree with their symbolic powers,
and that the ideal $Q(m)$ satisfies the persistence property for associated
primes. We also compute the analytic spread of $Q(m)$ in terms of the poset
$Q$.
