Is expiation required for an oath sworn on the Qur'an during childhood not to reveal a secret, while not remembering whether it was revealed or not, and if it is required, what is its expiation?

Whoever doubts whether they did something or not, the basic principle is that they did not do it until they are certain that they did. There is no expiation (kaffarah) upon you for this oath, because the basic principle is the acquittal of responsibility, and it is not mandated by doubt. It states in the Fiqh Encyclopedia that doubt concerning the origin of an oath or its breach does not obligate anything upon the doubter, because the basic principle is the acquittal of responsibility, and certainty is not removed by doubt.