Please follow the general troubleshooting steps first:

[Have you:]

• [Run] brew update and retried your prior step?
• [Run] brew doctor, fixed as many issues as possible[,] and retried your prior step?
• Confirmed this is problem with specific formulae and not Homebrew/brew? If it's a general Homebrew/brew problem[, then] please file this issue at https://github.com/Homebrew/brew/issues/new

Bug reports:

(Issue discussion continued/forked from here and here.)

For users who had both qt (Qt 4.x) and qt5 installed before the resolution and merging of #8306, running brew update to get the latest Homebrew implementation and tap changes on their systems and then running brew outdated -v to check for outdated packages results in the latter command reporting the existence outdated copies of Qt 5.x twice. I believe that this is a consequence of the fact that the same PR mentioned above made it so that qt is now an alias to qt5.

Output of brew config:

HOMEBREW_VERSION: 1.1.5
ORIGIN: https://github.com/Homebrew/brew.git
HEAD: 9cd5a21b473f0271b162bbe7f77f7d1468c0cfa1
Last commit: 2 weeks ago
Core tap ORIGIN: https://github.com/Homebrew/homebrew-core
Core tap HEAD: af7ffaddab5edea4ed2fd3011fbcfb310a8d80c8
Core tap last commit: 78 minutes ago
HOMEBREW_PREFIX: /usr/local
HOMEBREW_REPOSITORY: /usr/local/Homebrew
HOMEBREW_CELLAR: /usr/local/Cellar
HOMEBREW_BOTTLE_DOMAIN: https://homebrew.bintray.com
CPU: dual-core 64-bit core2
Homebrew Ruby: 2.0.0-p648
Clang: 8.0 build 800
Git: 2.11.0 => /usr/local/bin/git
Perl: /usr/local/bin/perl => /usr/local/Cellar/perl/5.24.0_1/bin/perl
Python: /usr/local/bin/python => /usr/local/Cellar/python/2.7.13/Frameworks/Python.framework/Versions/2.7/bin/python2.7
Ruby: /usr/local/bin/ruby => /usr/local/Cellar/ruby/2.4.0/bin/ruby
Java: 1.8.0_112, 1.8.0_111
macOS: 10.11.6-x86_64
Xcode: 8.2.1
CLT: 8.2.0.0.1.1480973914
X11: 2.7.11 => /opt/X11

Output of brew doctor:

Please note that these warnings are just used to help the Homebrew maintainers
with debugging if you file an issue. If everything you use Homebrew for is
working fine: please don't worry and just ignore them. Thanks!

Warning: You have unlinked kegs in your Cellar
Leaving kegs unlinked can lead to build-trouble and cause brews that depend on
those kegs to fail to run properly once built. Run `brew link` on these:
  qt5

(NOTE: I'd bet that this has to do with only one of the duplicate Homebrew-bookkept references to qt5. This is why I've left things as is for now, as I also suspect that this brew doctor notice will disappear as soon as said duplicate entries are fixed up.)

Output of brew outdated -v:

doxygen (1.8.12) < 1.8.13
gnutls (3.4.17) < 3.4.17_1
leptonica (1.73) < 1.74
qt5 (5.7.0) < 5.7.1
qt5 (5.7.0) < 5.7.1

(Notice the duplicate qt5 entries. Also, the other entries in this list have qt5 as a dependency — doxygen because I built it from source --with-qt5; gnutls because I built it from source --with-unbound, unbound depends_on libevent, and libevent depends_on doxygen by default, though this can be disabled using the --without-doxygen option despite the fact that using this option is neither existing nor contemplated practice on my part since I like having documentation stored locally on my machine; and leptonica because I built it from source --with-openjpeg, among other options, and openjpeg is a case identical to that of libevent in that it depends_on doxygen by default.)

(In addition, please finally note that I have not included any other logs in this issue report because the relevant problem manifests itself before I run brew upgrade --build-from-source, which has made me wary to do so, as Qt is expensive enough to do this once, let alone twice!)