Which technological advancement unify both the roman and the inca empires?

Ricky question for which both might be the answer. 
Given the size of both empires, one might want to say roads. This after all means more trade, easier migration and quicker movement of one's armies. This is certainly true of Rome. On the other hand, the Incas didn't have the wheel and so their idea of roads was little more than a series of tracks, and their bridges were made of grass and therefore temporary. 
Therefore, I would have to say the Astrolabe. The Incans worshipped the heavenly bodies, especially the sun, and seem to have been obsessed with predicting astronomical events. The Romans weren't opposed to such things either, and we know that they were well aware of such things as eclipses etc. and so, in a long-winded method I make my choice. 

The answer is 1) The Astrolabe.