Yesterday should be possessive. The style belongs to it. Likewise, The technology you're referring to belongs to tomorrow. Just because it isn't a person doesn't mean it can't have ownership of something.

As such the correct slogan would be:

YESTERDAY'S STYLE, TOMORROW'S TECHNOLOGY

Thanks.  This one ran & ran on various forums.  I was amazed at how serious these things are taken.  There are online forums just for this.  We concluded it was best to change the slogan.  Regardless of the correct form, enough people will think it is incorrect regardless of the one we use.  We have not got around to changing the web site yet but we will soon!

Thanks everyone!.  Arcol was the first to bring this up in a comment to a project log.  His comment above is also correct.  Just from the sample above, we can see that no matter how we deal with the phrase, a lot of people will think it is wrong.  As almost the only words on the opening page of a web site, having them (apparently) wrong is probably not a good strategy.  Back to the salt mines !

B) would be correct, given that you're using a common idiomatic form of "yesterday" that does not literally mean "the day before today".

In this form, it refers to "some time in the past", and that period could be defined as "the 50s and 60s".

Pulling out the "I write technical documentation for work" card :)

Better than the I'm a foreigner card Dave has !

"Yesterday's XXX, tomorrow's XXX" is a phrase, and as such it shouldn't be modified (by using plurals, for instance), because then it loses its special meaning and becomes gibberish. Also, whether you take "yesterday" to mean precisely the day before today, or the past in general, there is ever only one. You can't have multiple yesterdays. Either day or past, there is only one. Arguably you can have several futures, depending on which interpretation of the quantum theory you subscribe to, but that is a fairly recent idea that didn't get incorporated into the spoken language yet, so multiple tomorrows also sound awkward. The language only knows one past and one future.

The opposite of "classic" would be, in this case, "modern", I think.

This explanation seems the most logical to me, although (with explanation) either b) or c) point looks right to me.

But this stack page do show the original point: "a % of the audience will think it is incorrect."

So either find an "authoritive answer" from the neighbour university and stick to whatever solution they gave (perhaps an asterisk at the end of the slogan?:)

YESTERDAY'S STYLE, TOMORROW'S TECHNOLOGY*

*: confirmed by a study conducted by The University of Cambridge ;-)

Or some alternative suggestions:

a) Classic style equipped with future technology

b) Classic style equipped with futuristic technology

c) Future technology never looked so classic! 

Also sorry for ruining your slogan Maurice.

We can probably have different yesterdays too if we have parallel universes?  This is a good use of yesterdays in the plural form. There is an official plural of yesterday & it is yesterdays.  I wonder is understanding quantum theory as hard as figuring out how to use an (') correctly?  

The problem with multiple yesterdays is that in parallel universes those are different yourselves, all sharing some of their past. The present yourself will split into multiple yourselves, so you can say you have multiple futures (and each version of yourself will experience a different future), but you only have one past, and you share it with other versions of yourself from the parallel universes, who previously split off from your past self.

Unlike in physics, there is no "correct" in linguistics. There is only "common use" at a given time in a given population. The language is a living thing.

I'de trade all my tomorrows for a single yesterday.... (Kris Kristofferson)..  Gets complex in quantum & parallel. Probably a bad trade?. Unfortunately, Physics does not only deal in pure fact, it also uses convention.  In Physics (occasionally) it is generally accepted as correct until someone proves it is not correct, I suspect especially in the quantum world...Any topic has to be better than talking about an (')

isn't it only 's when it's short for is? You could always say "STYLE of YESTERDAY, TECHNOLOGY of TOMORROW"

Nope, possessive does the same.

I'm gladly pulling the "I'm a foreigner" wildcard :)

This seems like the most important thing.
I have no clue either, but it does seem like it's the right thing assuming yesterday and tomorrow are both possessive singular nouns.