Bash history is stored in ~/.bash_history. When you log out of the current shell, the shell’s history is written to the hidden file.

To empty out history, first clear the history from the current session

# history --help
# -c        clear the history list by deleting all of the entries
history -c

Then write the current history to the file

# -w        write the current history to the history file
history -w

Now if you look at the contents of ~/.bash_history, it will be empty, and no commands are available by pushing the up arrow.

cat ~/.bash_history

Combine it all together

history -c && history -w