Ternary itself, no.
But some other revolutionary improvement that necessitates a ternary computer, yes.
Possible candidates:
Optical computing - high speed
More accurately; low latency optical computing.
We transmit data via optical cables today, but we still do all our processing electronically. Converting photons to electric signals, doing something clever to make useful electric signals, converting them to photons and sending them on their. This takes time. Not much, but it happens often enough to be an annoyance.
Doing calculations on the photons themselves reduces the processing-delay.
Optical computing - massively parallel
One of the current big changes in computing today is the push from sequentially doing things (very quickly) to doing things simultaneously (even if it's slower).
When you run a silicon circuit really quickly it'll struggle to switch on/off in time(1). Typically, you solve this by turning up the voltage to make them reach switch voltages faster, but this means they get hot and melt.
But designing circuits that do many things at once is hard. We've been working on it for decades and we're still not very good at it. The first person to be good at it is going to be very rich indeed.
A combination of the above - large-scale neural networks
The discrete set {-1, 0, +1} looks a bit like the range [-1, +1]. That's the range of the Sigmoid Function. The Sigmoid is often used in artificial neural networks because of its similarities to the behaviour of real neurons.
This doesn't automatically mean you have Artificial General Intelligences running about the place. This doesn't necessitate Skynet or the Robot Apocalypse. Consider that the software. What your character has developed is the hardware.
Things that are more reasonable uses for a new, powerful, ANN; self-driving cars, market-trading software, better weather prediction (you'd be surprised how much money you could make by accurately predicting something that's assumed to be chaotic).
(1) I'd provide a citation for this, but my Google-Fu is a bit lacking today. I learned this while at university.