According to the topic of deformation quantization, the first few entries in the dictionary between
$$ \text{Quantum Mechanics}\quad\longleftrightarrow\quad\text{Classical Mechanics}\tag{0}$$
read
$$ \text{Operator}\quad\hat{f}\quad\longleftrightarrow\quad\text{Function/Symbol}\quad f,\tag{1}$$
$$ \text{Composition}\quad\hat{f}\circ\hat{g} \quad\longleftrightarrow\quad\text{Star product}\quad f\star g ,\tag{2}$$
and
$$ \text{Commutator}\quad [\hat{f},\hat{g}] \quad\longleftrightarrow\quad\text{Poisson bracket}\quad i\hbar\{f,g\}_{PB} + \color{red}{{\cal O}(\hbar^2)}. \tag{3}$$
Note that the correspondence (0) depends on which symbols one uses, e.g. Weyl symbols, and that there could in general be higher-order quantum corrections $\color{red}{{\cal O}(\hbar^2)}$ in the identification (3).
Example 1: (Fundamental CCR)
$$ [\hat{q},\hat{p}]~=~i\hbar{\bf 1}\quad\longleftrightarrow\quad
\{q,p\}_{PB}~=~1.\tag{4} $$
Example 2:
$$ [\hat{q}^2,\hat{p}^2]~=~4[\hat{q},\hat{p}]~ (\hat{q}\hat{p})_W\quad\longleftrightarrow\quad
\{q^2,p^2\}_{PB}~=~4\{q,p\}_{PB} ~qp, \tag{5}$$
where $(\ldots)_W$ stands for Weyl-symmetrization of operators. See also e.g. this Phys.SE post.
Example 3:
$$ [\hat{q}^3,\hat{p}^3]~=~9[\hat{q},\hat{p}]~ (\hat{q}^2\hat{p}^2)_W + \color{red}{\frac{3}{2}[\hat{q},\hat{p}]^3}\quad\longleftrightarrow\quad
\{q^3,p^3\}_{PB}~=~9\{q,p\}_{PB}~ q^2p^2.\tag{6} $$
Note that there are higher-order quantum corrections $\color{red}{{\cal O}(\hbar^3)}$ in eq. (6) even after Weyl-symmetrization.