8. As shown in Figure 2, in
rhombus $A B C D$, $\angle A B C$
$$
=120^{\circ}, B C=6 \sqrt{3} \text {, }
$$
$P$ is a moving point on the extension of $B C$
away from point $C$,
$A P$ intersects $C D$ at point $E$, and $B E$ is extended to intersect $D P$ at point
$Q$. When the moving point $P$ is at its initial position, $\angle Q B P=15^{\circ}$; at its final
position, $\angle Q B P=35^{\circ}$. Determine the length of the curve segment traced by point $Q$ as point $P$ moves.
8. As shown in Figure 6, connect $B D$, and construct the circumcircle of $\triangle A B D$ intersecting $A P$ at point $F$. Connect $B F$, $D F$, $F C$, and $C Q$. It is easy to see that
$$
\begin{array}{l}
\angle D F B=\angle D F P=\angle B F P=120^{\circ}, \\
\angle B F E=\angle E C P=120^{\circ}.
\end{array}
$$

Therefore, points $B$, $F$, $E$, and $C$ are concyclic.
Thus, $\angle E B C=\angle P F C$.
Since $\angle D F P=\angle D C P=120^{\circ}$,
$\Rightarrow$ points $D$, $F$, $C$, and $P$ are concyclic
$\Rightarrow \angle P F C=\angle C D P \Rightarrow \angle Q B P=\angle P F C$
$\Rightarrow$ points $B$, $C$, $Q$, and $D$ are concyclic
$\Rightarrow$ point $Q$ lies on the circumcircle of $\triangle B C D$.
It is easy to see that when point $P$ moves outward from point $C$ along the extension of $B C$, point $Q$ moves counterclockwise from point $C$ along the arc $\overparen{C D}$ of the circumcircle of $\triangle B C D$.

Since $\triangle B C D$ is an equilateral triangle with side length $6 \sqrt{3}$, its circumradius is $6 \sqrt{3} \times \frac{\sqrt{3}}{2} \times \frac{2}{3}=6$, so the circumference of its circumcircle is $2 \pi \times 6=12 \pi$.

Since the central angle corresponding to the arc $\overparen{Q_{1} Q_{2}}$ is $35^{\circ}-15^{\circ}=20^{\circ}$, the arc length of $\overparen{Q_{1} Q_{2}}$ is $40^{\circ}$, which is $\frac{1}{9}$ of the entire circumference. Therefore, when point $P$ moves, the length of the curve $\overparen{Q_{1} Q_{2}}$ that point $Q$ travels is $\frac{12 \pi}{9}=\frac{4 \pi}{3}$ (as shown in Figure 7).