Queen Elizabeth II, as the Head of State, has to be neutral in all political matters. This means she cannot vote or take a stand on any election. However, many royal fans are wondering if members of the British Royal family can enter politics, specifically, run for the Prime Minister post. As it happens, it's possible. 

In a discussion site called, Quora, a lot of people cannot help but wonder if Prince William or Prince Harry can run for UK's top post in the government. Based on the answers and statements given, it seemingly appears that there is no direct and clear answer if this can truly happen. 

Some fans are convinced that Prince William and Prince Harry are eligible to run for the Prime Minister's post. This is because they pass all the requirements and qualifications needed to run for the position. 

To become UK's Prime Minister, there are only two main qualifications: 1) the aspirant should be over 18 years of age; and 2) he or she must be a British citizen or a citizen of a Commonwealth country, or a citizen of the Republic of Ireland. Technically, these all apply to Prince William and Prince Harry's standing. But, as per reports, a royal member wishing to run for the post should first convince two Liberal Democrats and, of course, Queen Elizabeth. 

While this perspective is quite viable, there are some royal followers who believe otherwise. Based on the other answers shared on the site, Prince William cannot run for the Prime Minister post because he is "realistically" going to be a king, being he's the second person in line for the throne. 

The case is different, however, for Prince Harry considering that he'll only become king, when Prince Charles, Prince William, Prince George, Princess Charlotte, and Prince Louis couldn't or wouldn't sit to the throne. This is also asserted by Express UK as the publication believes that Prince Harry can become UK's Prime Minister if he wishes to. 

Nothing can stop the Duke of Sussex from running as Prime Minister, especially if Queen Elizabeth will give her blessings. After "ticking" all the boxes of the qualifications list, all he needs to do is to have a party that will nominate him. But, if none, he can run independently.

All in all, Prince Harry is seemingly much more viable to run for the post than Prince William. However, many sources and experts claim that he is not a political type. Therefore, it's very unlikely that he'll run for the post and become UK's Prime Minister.