So going of of this chart http://imgur.com/tT700Fu The Romans technologically got as far as the end of the Renaissance. (Comparison Drawn here http://imgur.com/AtEl1HS ) And During the renaissance is when the gun or "hand cannon" began to be produced in Europe... Did the Romans have some sort of ancient (or I guess not so ancient if they were on par with the Renaissance) firearm?
The simple answer is: the Romans didn't have gunpowder, therefore they didn't have firearms. Gunpowder was invented in China, probably in the 9th century, and spread to Europe in the 13th century.
The longer answer... That diagram is very misleading. You could get some interesting responses from some experts on science & technology, ancient history, and the renaissance, if you started a new thread asking about this.
To pick up just a few issues with the diagram:
What's it actually measuring, and how? It seems to be showing the rate of scientific advancement. This is not an easy thing to measure. It also means that the diagram does not show that medieval science & technology was at the same level as ancient Egyptians (the Egyptians couldn't build clocks) or that the Romans had the same technology as Europe at the end of the renaissance.
Why are the specific regions picked out? Ancient Egypt was pretty backward in many areas of science and technology. Why pick out the "Christian dark ages" from 500-1500(?) CE rather than the flourishing of Islamic scholarship or Chinese science & technology for parts of this period? What about Orthodox Christianity, which didn't have this "dark age"?
The idea that nothing useful was discovered or invented in Europe during the dark ages or medieval period is generally rejected now. The term "dark ages" seems to have largely fallen out of use. To take some random examples of technologies that were invented in or introduced to Europe, there's the stirrup, gothic arches, the water wheel, gunpowder & firearms, mechanical clocks, great advances in shipbuilding & navigation, the introduction of arabic numerals and of zero...
That ambiguously labeled chart, with no description as to what exactly is measured, nor how it is measured, and generously providing zero references, rears its literally ugly head on this sub periodically.
Please help yourself to the FAQ section, linked on the right hand side, for the many frequently asked questions related to the dark ages and the fallacies of that term.