What time period lasted for more than half of the hypothesized history of the Earth?

The best answer to this would be Proterozoic (although it actually lasted a little less than half by estimates, but it's the longest period.)

It started around 2500 million years ago and lasted till 500 million years ago - 2000 million years long. The estimated age of Earth is 4500 million years old - a little over twice this period