Count the new element added and removed from the previous group from a dataframe

Not sure about the speed implications, but a big part of this is essentially comparing consecutive counts, so diff came to mind.

tab <- table(df$Quarter, df$Name)
tab <- rbind(tab[1,,drop=FALSE], diff(tab))
out <- rbind(added = rowSums(tab == 1), left = rowSums(tab == -1))

#      Q1 2019 Q2 2019 Q3 2019 Q4 2019
#added       3       2       0       0
#left        0       1       0       2

If you need the long output specifically:

setNames(data.frame(as.table(out)), c("status","quarter","count"))
#  status quarter count
#1  added Q1 2019     3
#2   left Q1 2019     0
#3  added Q2 2019     2
#4   left Q2 2019     1
#5  added Q3 2019     0
#6   left Q3 2019     0
#7  added Q4 2019     0
#8   left Q4 2019     2

Split to create a list and map the two lists to get the length of the 'uneaual' elements, i.e.

l1 <- split(df$Name, df$Quarter)
do.call(rbind, Map(function(x, y) { i1 <- length(setdiff(x, y)); 
                                    i2 <- length(setdiff(y, x)); 
                                    data.frame(Added = i1, Left = i2)},
          l1[-1], l1[-length(l1)]))

#        Added Left
#Q2 2019     2    1
#Q3 2019     0    0
#Q4 2019     0    2

You can tidy the output the way you want

The following works by first turning the Name column into a list of names by Quarter, and then comparing every Quarter with the previous Quarter using purrr::map2_int. Finally, the two columns that were added, Added and Left are pivoted into long form using tidyr::pivot_longer.

library(tidyverse)

df %>%
  group_by(Quarter) %>%
  summarise(names = list(Name)) %>%
  mutate(Added = map2_int(names, lag(names, default = list(list())), ~ length(setdiff(.x, .y))),
         Left = map2_int(names, lag(names, default = list(list())), ~ length(setdiff(.y, .x)))) %>%
  pivot_longer(Added:Left, names_to = "status", values_to = "Count") %>%
  select(-names)

Result:

# A tibble: 8 x 3
  Quarter status Count
  <chr>   <chr>  <int>
1 Q1 2019 Added      3
2 Q1 2019 Left       0
3 Q2 2019 Added      2
4 Q2 2019 Left       1
5 Q3 2019 Added      0
6 Q3 2019 Left       0
7 Q4 2019 Added      0
8 Q4 2019 Left       2

Here is one way keeping the data wide. We split the data into list of dataframes based on Quarter. Using map2 we compare one quarter and the next quarter values and count number of people added and left in each quarter. Calculate the values for first quarter separately and bind it to the original dataframe.

library(tidyverse)

list_df <- df %>% group_split(Quarter)

list_df %>%
   .[[1]] %>%
  summarise(quarterYear  = first(Quarter),
            status = c('Added', 'Left'), 
            Count = c(n(), 0)) %>%
    bind_rows(map2_df(list_df[-1], list_df[-length(list_df)], 
            ~tibble(quarterYear = .x$Quarter[1],
                    status = c('Added', 'Left'), 
                    Count = c(sum(!.x$Name %in% .y$Name), 
                              sum(!.y$Name %in% .x$Name)))))


# A tibble: 8 x 3
#  quarterYear status Count
#  <chr>       <chr>  <dbl>
#1 Q1 2019     Added      3
#2 Q1 2019     Left       0
#3 Q2 2019     Added      2
#4 Q2 2019     Left       1
#5 Q3 2019     Added      0
#6 Q3 2019     Left       0
#7 Q4 2019     Added      0
#8 Q4 2019     Left       2

Using the same logic in base R:

list_df <- split(df, df$Quarter)
temp <- list_df[[1]]

rbind(data.frame(quarterYear = temp$Quarter[1], 
                 status =  c('Added', 'Left'), 
                 Count = c(nrow(temp), 0)),
   do.call(rbind, Map(function(x, y) 
      data.frame(quarterYear = x$Quarter[1],
                 status = c('Added', 'Left'), 
                 Count = c(sum(!x$Name %in% y$Name), sum(!y$Name %in% x$Name))), 
list_df[-1],list_df[-length(list_df)])))