LocostUSA.com

I did autocross with my Locost last year, but was getting some nasty rear wheel hop when getting on the power coming out of corners. I didn't build the car, but it has ATV shocks/springs in the rear and the factory '85 Corolla GTS rear end. I'm thinking maybe the springs are too soft? Any thoughts?

I expect the damping (not the springs) are too soft. You may be able to fit an adjustable seat and damping coilover to the existing atv attachments. Remove one side, raise the axle to the full bump position, then measure the distance between the mount eye centers on the axle and chassis. Short enough is more critical than length. Worst case, the attachments can be altered for a longer compressed length coilover. If the atv springs won't fit the new coilover, measure the atv spring rate using a bath scale and a press to squeeze it down at least an inch or measure the spring and do the math. Seat adjustment and distance at full bump will help determine the new spring off-the-shelf length.

I'd also check the condition (loose and/or soft?) of the link ends. You can fit a camera under the car to record wheel movement on hard accel to see how much the link attachments allow the axle to oscillate.
It looks like a good, parallel, equal length links that are level at ride height with bushing links on a fixed axle attachment (versus a "birdcage"). For more anti-squat/traction options, I'd consider reworking the chassis upper link attachment plates with multiple holes for a shorter upper link and a slightly lower mounting hole option, keeping the lower links level at ride height and as long as possible. Also making new links that are adjustable length and spherical ends with room for clearance spacers on each side of the spherical (may need to adjust distance between double shear bracket flanges).

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Miata UBJ: ES-2074R('70s maz pickup)
Ford IFS viewtopic.php?f=5&t=13225&p=134742
Simple Spring select viewtopic.php?f=5&t=11815
LxWxHt
360LA 442E: 134.5x46x15
Lotus7:115x39x7.25
Tiger Avon:114x40x13.3-12.6
Champion/Book:114x42x11
Gibbs/Haynes:122x42x14
VoDou:113x44x14
McSorley 442:122x46x14
Collins 241:127x46x12

I have the same set-up [an 87 GTS axle and zero wheel hop] Rear spring rate of 170 lbs with GAZ shocks set at 13 clicks (out of 21) the upper trailing arm front frame connection is set 1 inch lower for anti-squat, Vs parallel arms. Lowering the top trailing arm connect may be a problem unless you move your brake calipers to the rear of the axle. Moving the calipers to the rear of the axles does require pulling the axles to flip the brake caliper mounting brackets from side to side. My arms are 11.5" length design with a rubber bushing on the frame connect side and rod ends on the axle connection. I also lower the diff pinon angle downward at .5 degrees. If you have any other questions, you can send me a PM.

Also looks like the atv coilovers attachments to the chassis are much further apart than the axle end attachments, for a falling rate. The shortest compressed length gaz with 170# and a new axle end coilover mount, hanging off behind the trailing arm brkt, would probably be a big improvement.

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Miata UBJ: ES-2074R('70s maz pickup)
Ford IFS viewtopic.php?f=5&t=13225&p=134742
Simple Spring select viewtopic.php?f=5&t=11815
LxWxHt
360LA 442E: 134.5x46x15
Lotus7:115x39x7.25
Tiger Avon:114x40x13.3-12.6
Champion/Book:114x42x11
Gibbs/Haynes:122x42x14
VoDou:113x44x14
McSorley 442:122x46x14
Collins 241:127x46x12

My Locost also has a Corolla GTS solid axle. Gaz coilovers, 150 lb. springs. I installed a 1.5 way KAAZ limited slip which is awesome for autocross. I swapped my brakes side to side and mounted them at the rear as well.
I think a limited slip may reduce the chance of wheel hop, but maybe your springs are too soft.

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book 4AGE build, Supra W58 transmission, Corolla GTS live axle