Bitcore - The Currency Of The People
Bitcore - The Pink Future

But "Bitcore - Currency Of The People" should be enough, no need for a "the"

Personally I don't like a slogan that tells people what something is NOT or something do NOT do - it should give a glimpse of what it stands for and not the other way around.
If it comes to cryptocurrency it isn't that revolutionary to NOT fork and the slogan may be outdated in some months.

Another bad name would be "Moon Lambo Moon Lambo" or any other reference to moons in this context of cryptocurrencies - couldn't sound more suspicious for a project who wants to establish itsself in longterm.

If it will be for and "of the people" a latin name doesn't sound that great and wellknown, does it?

I think big brands have something in common for a reason - their slogans are simple and understandable most of the time.

Sorry for the critics.
There wasn't any suggestions from my side so far, so: Good job!