Speaker wire size VS speaker size. Should bigger speakers use bigger wires?

Its the purity of wire (i like copper)  and the design that make a wire great size is not as important although again it depends on the design.Good luck.

Based on what I read, wire size selection should be based on how low in impedance the speaker system goes.. My totem hawks are more listenable (more punchy bass) with belden 10awg than with kimber 4pr 14awg.  The opposite is true with my wharfedale denton.

My Infinity speakers can drop below the 4-ohm range from what I read quite a few years ago.  I guess their really large Kappa 9.1 series 2 dip way below, maybe even to the 2-ohm range.  The Infinities make an amplifier work that is for sure.  I cooked a Yamaha MX-1 and a Krell KSA 200s with them.  The 300cx makes them sing. 
-John

randy-11 has it right.  I would only add that paying attention to the conductor and dielectric quality as well as connectors on the cables can/do make a discernible difference.

One surprise for me was when I got my Vandersteen Treo CT speakers with their low 85db efficiency, my very heavy gauge "Earth series" cables from Audioquest made with lots and lots of very pure copper lent a mid bass "thickness" to the sound.  Moving to Paul Laudati's Clear Day silver cables opened up the sound, ridding my system of that thickness and revealing smooth treble detail that I had believed the carbon tweeters were capable of but I didn't know what I had been missing until inserting the Clear Days into the system.

One thing however, in support of randy-11's statement about efficiency, I had to buy the Clear Days with the most silver, the double shotguns to accommodate to the low efficiency.

Speaker wires are important, bur an overkill is surely not needed.
Depending  the need of a high bass level 0.75mm2 till 1,5mm2 is usefull the rest is overkill.
For the rest: 4-pole calculations will clear everything.
regards
Ton