Instead of scalebox you may use resizebox
\documentclass{article}
\usepackage{graphicx,amsmath}
\newcommand\bigfrown[2][\textstyle]{\ensuremath{%
\array[b]{c}\text{\resizebox{6ex}{.7ex}{$#1\frown$}}\\[-1.3ex]#1#2\endarray}}
\begin{document}
\bigfrown{PQR}
\end{document}
This is very manual and you have to adjust the parameters of resizebox by yourself.
Edit:
Atleast with this code you don't have to guess the width:
\documentclass{article}
\usepackage{graphicx,amsmath,calc}
\newlength{\mywidth}
\newcommand\bigfrown[2][\textstyle]{\ensuremath{%
\array[b]{c}\text{\resizebox{\mywidth}{.7ex}{$#1\frown$}}\\[-1.3ex]#1#2\endarray}}
\begin{document}
\settowidth{\mywidth}{PQRSPQ}
\bigfrown{PQRSPQ}
%\setlength{\mywidth}{0pt}
\settowidth{\mywidth}{PQR}
\bigfrown{PQR}
\end{document}
