We introduce and study noncommutative (or „quantized") versions of the algebras of holomorphic functions on the polydisk and on the ball in $\mathbb C^n$. Specifically, for each $q\in\mathbb C\setminus{ 0}$ we construct Fréchet algebras $\mathcal O\_q(\mathbb D^n)$ and $\mathcal O\_q(\mathbb B^n)$ such that for $q=1$ they are isomorphic to the algebras of holomorphic functions on the open polydisk $\mathbb D^n$ and on the open ball $\mathbb B^n$, respectively. In the case where $0 < q < 1$, we establish a relation between our holomorphic quantum ball algebra $\mathcal O\_q(\mathbb B^n)$ and L.L. Vaksman's algebra $C\_q(\bar{\mathbb B}^n)$ of continuous functions on the closed quantum ball. Finally, we show that $\mathcal O\_q(\mathbb D^n)$ and $\mathcal O\_q(\mathbb B^n)$ are not isomorphic provided that $|q|=1$ and $n\ge 2$. This result can be interpreted as a $q$-analog of Poincaré's theorem, which asserts that $\mathbb D^n$ and $\mathbb B^n$ are not biholomorphically equivalent unless $n=1$.