Answer:
Connect the microphone directly to the PS4 via USB-C (with the included USB-A adapter). Then go to the settings under "Devices" → "Audio devices" and select "USB headset (HyperX QuadCast)" for both the input and output device.

Step by step:
1st connection: Plug the USB cable (with adapter for USB-A port) into the PS4.
2. PS4 menu: Open "Settings" → "Devices" → "Audio devices".
3. setup:
- Input device: Select "USB headset (HyperX QuadCast)".
- Output device: Select "USB headset (HyperX QuadCast)".
- Audio output: Activate "All audio outputs".
4. headphones: Optionally, you can connect headphones to the microphone via the 3.5 mm output.

Note: According to the manufacturer, the QuadCast 2 is compatible with PS4/PS5. In case of problems, check that the cable is fully plugged in or restart the console.