4. Inside an irregular convex cube, there is a point $Q$. A line $l$ passing through $Q$ intersects the surface of the cube at points $M$ and $N$. The number of lines $l$ for which $Q M = Q N$ holds is $\qquad$.
4. Infinitely many.

Let $Q$ be any point, and let $T_{1}$ be any plane passing through $Q$ that intersects the surface of the cube in a closed curve $\Gamma_{1}$, with point $Q$ inside the curve.

Let $N_{0}$ be the point on curve $\Gamma_{1}$ that is closest to $Q$. Extend $N_{0} Q$ to intersect curve $\Gamma_{1}$ at point $M_{0}$. Let $N_{\alpha}$ be the point on curve $\Gamma_{1}$ that is reached by rotating $N_{0}$ clockwise around $Q$ by an angle $\alpha$, and let $N_{\alpha} Q$ intersect curve $\Gamma_{1}$ at point $M_{\alpha}$.
Let $f(\alpha)=Q N_{\alpha}-Q M_{\alpha}(\alpha \in[0, \pi])$.
Clearly, $f(\alpha)$ is a continuous function.
Also, $f(0) \leqslant 0, f(\pi) \geqslant 0$, so there exists $\alpha \in [0, \pi]$ such that $f(\alpha)=0$, meaning there is a line $l_{1}$ in plane $T_{1}$ that satisfies the condition.

Assume there are already $k$ lines $l_{1}, l_{2}, \cdots, l_{k}$ that satisfy the condition. Then, through point $Q$, draw any plane $T_{k+1}$ such that
$$
l_{i} \notin T_{k+1}(i=1,2, \cdots, k) \text {. }
$$

Similarly, it is known that there exists a line $l_{k+1} \in T_{k+1}$ that satisfies the condition.