A simple implementation of Google+ Sign-In on iOS using version 1.3.0 of the SDK: http://developers.google.com/+/mobile/ios

AppDelegate.m

//
//  AppDelegate.m

#import "AppDelegate.h"
#import "GoogleOpenSource/GTMOAuth2Authentication.h"
#import "MainViewController.h"
#import "SignInViewController.h"

@implementation AppDelegate

- (BOOL)application:(UIApplication *)application
    didFinishLaunchingWithOptions:(NSDictionary *)launchOptions {
  self.window = [[UIWindow alloc] initWithFrame:[[UIScreen mainScreen] bounds]];

  // Before configuring, make sure to have:
  // 1. Created a custom URL matching your bundle ID.
  // 2. Added the GooglePlus.framework, GoogleOpenSource.framework
  //    and GooglePlus.bundle to your project.
  // 3. Added the SystemConfiguration and Security frameworks.
  // 4. Make sure you have -ObjC in your Other Linker Flags setting. Otherwise,
  //    what is likely to happen is that the sign in button wont be able to find
  //    the categories it needs, and you'll get odd errors like:
  //    "[__NSDictionaryM gtm_httpArgumentsString]: unrecognized selector".
  //    The casing is important!

  // Configure the client ID from the API console:
  // http://developers.google.com/console
  [[GPPSignIn sharedInstance] setClientID:@"YOUR_CLIENT_ID"];

  // Configure callbacks from the sign-in process to be sent to this object.
  // Note that we added the GPPSignInDelegate protocol to the AppDelegate.h.
  // If we don't set this, then the process will complete, but we'll never get a
  // call to our |finishedWithAuth:error:| method!
  [[GPPSignIn sharedInstance] setDelegate:self];

  // If we needed to request access to write app activities, we could also set
  // the |actions| property. If we wanted to add any scopes, we would add to the
  // scopes property. In the absence of more specific requirements, the default
  // if for the Google+ login scope.  

  // Now we can try and sign-in right away. We could store the return type in a
  // property in case the library requires making a network call and the
  // callback is delayed - this allows controllers to check whether a callback
  // can be expected. 
  if (![[GPPSignIn sharedInstance] trySilentAuthentication]) {
    // If we aren't silently signing-in, we'll might show a sign-in button
    // If we are, we'll defer the decision till the |finishedWithAuth:error:|
    // call so we don't have a flash of the sign-in button. 
    self.viewController = [[SignInViewController alloc] init];
  } else {
    // Create the view controller that the user will interact with. We don't
    // need to pass anything over here, as it can check the state of play using
    // the |GPPSignIn sharedInstance|.
    self.viewController = [[MainViewController alloc] init];
  }

  self.window.rootViewController = self.viewController;
  [self.window makeKeyAndVisible];
  return YES;
}

- (void)finishedWithAuth:(GTMOAuth2Authentication *)auth
                   error:(NSError *)error {
  if (error) {
    // If the error value is set, something has gone wrong. There are several
    // possible values, but all indicate the user is not signed-in, and the auth
    // should not be used.
    NSLog(@"Sign-In Error: %@", error);
    if ([error code] == -1) {
      NSLog(@"Unknown error, but user probably cancelled.");
    } else if([error code] == 400 &&
              [[error domain] isEqualToString:@"com.google.HTTPStatus"]) {
      NSLog(@"400 error, user has probably disconnected using the "
            "app managment page.");
    }
    // Ensure we have a clear setup for next sign in.
    [[GPPSignIn sharedInstance] signOut];
  } else {
    // We're signed in! We wont do much here, as the controllers can test
    // the sign-in state via the |sharedInstance| on |GPPSignIn|, but for more
    // complex applications writing an notification might be a good way of
    // signalling the state change.
  }
}

- (BOOL)application:(UIApplication *)application
            openURL:(NSURL *)url
  sourceApplication:(NSString *)sourceApplication
         annotation:(id)annotation {
  // We need to use the helper GPPURLHandler class to manage the open URL. This
  // is used for the response from the Google+ application or the browser after
  // a user has been taken away to authorise an application. It also handles
  // callbacks after sharing, and if the application is opened with a deeplink!
  //
  // If you need to combine this with other URL handlers, you can just test the
  // returned BOOL, and hand over to the next handler in the chain if the return
  // is NO.
  return [GPPURLHandler handleURL:url
                sourceApplication:sourceApplication
                       annotation:annotation];
}

@end

SignInViewController.m

//
//  ViewController.m

#import "GooglePlus/GooglePlus.h"
#import "SignInViewController.h"

@implementation SignInViewController

- (void)viewDidLoad {
  [super viewDidLoad];

  // Create the Google+ Sign-In button. It automatically refers to the
  // |GPPSignIn sharedInstance| so we don't need to wire it up.
  GPPSignInButton *button = [[GPPSignInButton alloc] init];

  // We can configure various properties, in this case, choosing the wide
  // version of the button. The image and translated text are loaded from the
  // GooglePlus.bundle - if you forget that, the button will be there, but
  // invisible!
  [button setStyle:kGPPSignInButtonStyleWide];

  [self.view addSubview:button];

  // This is just the autolayout code to center our button.
  button.translatesAutoresizingMaskIntoConstraints = NO;
  
  // Center the button vertically.
  NSLayoutConstraint *constraint = [NSLayoutConstraint
                                    constraintWithItem:button
                                    attribute:NSLayoutAttributeCenterY
                                    relatedBy:NSLayoutRelationEqual
                                    toItem:self.view
                                    attribute:NSLayoutAttributeCenterY
                                    multiplier:1.0f
                                    constant:0.0f];
  [self.view addConstraint:constraint];

  // Center the button horizontally.
  constraint = [NSLayoutConstraint
                constraintWithItem:button
                attribute:NSLayoutAttributeCenterX
                relatedBy:NSLayoutRelationEqual
                toItem:self.view
                attribute:NSLayoutAttributeCenterX
                multiplier:1.0f
                constant:0.0f];
  [self.view addConstraint:constraint];
}

@end