No, it is not possible to fit an infinite number of objects, such as play balls, inside a finite-sized box, regardless of the technology used, including quantum technology. The concept of infinity implies an unbounded quantity that cannot be contained within a finite space.
Quantum technology, including quantum computing and quantum information processing, offers unique computational capabilities but does not defy the fundamental principles of physics and mathematics. While quantum systems can exhibit superposition and entanglement, allowing for novel computational possibilities, they do not enable the violation of basic spatial constraints.
Even with advanced quantum technology, the size and capacity of a physical object, such as a box, remain finite. While quantum algorithms can potentially solve complex problems more efficiently than classical computers, they do not alter the inherent limitations of physical space.
Therefore, the idea of fitting an infinite number of objects within a finite-sized box, whether with classical or quantum technology, is not feasible.