A secret organization recruits genius children, and trains them to become future politicians. After 15 years, they will hold positions for THE NEW REPUBLIC OF THE UNITED STATES. The cult believes that their plan will ensure the survival of the American Empire, as the supreme world power for the next 500 years.

OK, so, based off what you have submitted, it's too vague, no discernible protagonist.  No discernible conflict.  And honestly, it's doesn't entice me to want to open the script.

Ahem... the primary function of a logline is to serve as a window to your screenplay. The logline is the succinct essence of the critical elements of the story that must be expressed to get a clear picture of the throughline of your story.

At the moment, yours need some windex.  In other words, it needs to be less clunky, and more focused.  

All that aside, I'll offer a suggestion that maybe you can modify and build off of as you see fit, but first I need more help from you:  

Have you read any articles or tips on how to construct loglines?  

Who is the main protagonist and what does he or she want?
Who (antagonist) or what is standing in the way of the main character?
What are the stakes?
What makes this story unique?

I agree with Ghostwriter and believe you can start by just shortening this:

Quoted Text A secret organization recruits genius children, and trains them to become future politicians. After 15 years, they will hold positions for THE NEW REPUBLIC OF THE UNITED STATES. The cult believes that their plan will ensure the survival of the American Empire, as the supreme world power for the next 500 years

To this:

A secret organization recruits and trains genius children to prepare them to implement their vision of a new and permanent American empire.

That'll give you a little more room to address the issues in the above post