Some people reported problems with cutechess and chessx because of different versions of qt.

% brew info qt     
qt: stable 6.0.2 (bottled), HEAD
% brew info qt@5
qt@5: stable 5.15.2 (bottled) [keg-only]

The latter one is important. Check whether qt@5 is installed. Otherwise install it with:

brew install qt@5

Here are new binaries for the M1 (arm64) which should work now:

• chessx-1.5.6.app
  Source: https://github.com/Isarhamster/chessx/releases
• Cutechess-1.2.app
  cutechess-cli-1.2
  Source: https://github.com/cutechess/cutechess/releases/tag/1.2.0

And a small cook book if you want to build these qt based applications for your own. First you have to set some paths for successfully compiling your software:

% ls -l /opt/homebrew/Cellar/qt@5 
total 0
drwxr-xr-x  23 user  admin  736  9 Mär 22:52 5.15.2
% export PATH=/opt/homebrew/Cellar/qt@5/5.15.2/bin:$PATH
% export LDFLAGS=-L/opt/homebrew/Cellar/qt@5/5.15.2/lib 
% export CPPFLAGS=-I/opt/homebrew/Cellar/qt@5/5.15.2/include 
% export PKG_CONFIG_PATH=/opt/homebrew/Cellar/qt@5/5.15.2/lib/pkgconfig

Switch to your source of cutechess and do:

% cd ~/local/src/cutechess-1.2.0
% qmake
% make
% ls -l projects/gui/
...
drwxr-xr-x   3 user  staff       96 17 Mär 10:31 Cute Chess.app
...
% ls -l projects/cli/
...
-rwxr-xr-x   1 user  staff  1200794 17 Mär 10:32 cutechess-cli
...

For chessx-1.5.6 it’s practically the same:

cd ~/local/src/chessx-1.5.6-lw/
% qmake
% make
% ls -l release/
drwxr-xr-x  3 user  staff    96B 16 Mär 13:30 chessx.app