There are certainly numerous developments and inventions from these time periods which you could choose. If you must limit it to just two, I would suggest focusing on the development of writing and the invention of the wheel.
The earliest known form of writing developed in Mesopotamia during the Bronze Age. When the Sumerians invented cuneiform writing on clay tablets, they changed forever how people communicate and keep records. Early on, writing took on many of the same purposes as it does today. It was used to record important events, keep track of business transactions, record laws, send messages over distances, and put stories into writing. This shaped the development of various civilizations in uncountable ways.
Throughout the ancient world, different civilizations adapted and created their own forms of writing. Not only did this allow for the further development of the arts and facilitate commerce and law, but it also impacted society. Throughout the entire ancient period, only a minority of people ever became literate. This disparity may have led to a greater social divide as the literate class was able to use this ability to exert themselves more in the political and economic spheres.
Another development of the ancient world worth considering is the invention of the wheel. The earliest known wheels were used to make pottery in Bronze Age Mesopotamia. Over time they came to be used for transportation. Wheels allowed people to work more efficiently as they moved heavy loads over land. This improved agriculture and commerce as goods and food could be brought to markets more quickly and efficiently. Heavy stone wheels were used to crush grain, fruits, and seeds, making it easier to mass-produce flours and oils. As such, the wheel may be partially responsible for the growth in population during the Bronze Age and the spread of agriculture. Wheels were also used on the battlefield. Wheeled chariots were the favored shock weapons of several armies, such as the Hittites and the Persians, helping them conquer vast territories.