I have been trying to build an antenna for experimenting with GPS reception and rtl-sdr for a couple of weeks now but I had previously failed at building any working helical or quadrifilar helical antenna for it.
This evening I gave it a go once more after stumbling across a design called a skew planar wheel antenna when reading through /r/RTLSDR where someone in this thread asked about an antenna for receiving NOAA weather satellite images.
I had known about a similar type, the clover leaf antenna because RC hobbyists use them for their FPV systems and I have seen those before on YouTube.
The skew planar wheel antenna is a circularly polarised omnidirectional antenna, exactly what you want for satellite reception. It might not be as optimal as a QFH but it is definitely easier to construct even when you are in a pinch and only have gardening wire available, see image above.
Please note that this antenna looks like a short at DC. If your GPS is set up to provide power to an LNA/active antenna make sure to disable it, otherwise it will be shorted to ground.
I have taken the design details (measurements) from here
The elements are each slightly longer than the free space wavelength of the frequency of interest. The formula for the element length in centimeters is l = 31329 / F.
I do not know how well this antenna performs as I lack test equipment for taking any meaningful measurements at the GPS L1 frequency.
As you can see in the screenshot, I do receive something. Wether that is part of the GPS CDMA spread spectrum transmission I cannot yet tell. All I know is that I can receive this with the homebrew antenna and I do not receive anything with a commercial passive ceramic patch GPS antenna under the same circumstances. The only amplification used here is the LNA of the rtl-sdr dongle. I already have designed and ordered PCBs for a dedicated GPS LNA and filter IC.
I have tested this antenna with a commercial Sparkfun Venus GPS breakout board and (after removing the LNA biasing inductor) I was able to indeed acquire a GPS fix using this antenna.