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The advantages of using ternary logic vs binary logic to do our computing would be that we can hold the same amount of information in less memory, and we can process more information at once.

To implement the hardware based on ternary logic is much more complex and that is why binary won. It is much easier to process and store 2 representational states in each bit then 3. It is easier to tell if there is a charge or not vs determining the difference between a high charge, a medium charge, or no charge. It is easier to store and read information with 2 polarities vs 3 polarities.